Covering properties and square principles

Covering matrices were used by Viale in his proof that the Singular Cardinals Hypothesis follows from the Proper Forcing Axiom and later by Sharon and Viale to investigate the impact of stationary reflection on the approachability ideal. In the course of this work, they isolated two reflection principles, CP and S, which may hold of covering matrices. In this paper, we continue previous work of the author investigating connections between failures of CP and S and variations on Jensen’s square principle. We prove that, for a regular cardinal λ > ω ₁, assuming large cardinals, □(λ, 2) is consistent with CP(λ, θ) for all θ with θ ⁺ < λ. We demonstrate how to force nice θ-covering matrices for λ which fail to satisfy CP and S. We investigate normal covering matrices, showing that, for a regular uncountable κ, □ _κ implies the existence of a normal ω-covering matrix for κ ⁺ but that cardinal arithmetic imposes limits on the existence of a normal θ-covering matrix for κ ⁺ when θ is uncountable. We introduce the notion of a good point for a covering matrix, in analogy with good points in PCF-theoretic scales. We develop the basic theory of these good points and use this to prove some non-existence results about covering matrices. Finally, we investigate certain increasing sequences of functions which arise from covering matrices and from PCF-theoretic considerations and show that a stationary reflection hypothesis places limits on the behavior of these sequences.     

Model-theoretic applications of cofinality spectrum problems

We apply the recently developed technology of cofinality spectrum problems to prove a range of theorems in model theory. First, we prove that any model of Peano arithmetic is λ-saturated iff it has cofinality ≥ λ and the underlying order has no (κ, κ)-gaps for regular κ < λ. We also answer a question about balanced pairs of models of PA. Second, assuming instances of GCH, we prove that SOP ₂ characterizes maximality in the interpretability order ?*, settling a prior conjecture and proving that SOP ₂ is a real dividing line. Third, we establish the beginnings of a structure theory for NSOP ₂, proving that NSOP ₂ can be characterized by the existence of few so-called higher formulas. In the course of the paper, we show that p_s = t_s in any weak cofinality spectrum problem closed under exponentiation (naturally defined). We also prove that the local versions of these cardinals need not coincide, even in cofinality spectrum problems arising from Peano arithmetic.     

Quotients and colimits of κ-quantales

Let κQnt be the category of κ-quantales, quantales closed under κ-joins in which the monoid identity is the largest element. (κ is an infinite regular cardinal.) Although the lack of lattice completeness in this setting would seem to mitigate against the techniques which lend themselves so readily to the calculation of frame quotients, we show how to easily compute κQnt quotients by applying generalizations of the frame techniques to suitable extensions of this category.The second major tool in the analysis is the free κ-quantale over a λ-quantale, κ?λ. Surprisingly, these can be characterized intrinsically, and the generating sub-κ-quantale can even be identified. The result that the λ-free κ-quantales coincide with the λ-coherent κ-quantales directly generalizes Madden?s corresponding result for κ-frames.These tools permit a direct and intuitive construction of κQnt colimits. We provide two applications: an intrinsic characterization of κQnt colimits, and of free (over sets) κ-quantales. The latter is a direct generalization of Whitman?s condition for distributive lattices.     

Duality in nondifferentiable multiobjective fractional programming problem with generalized invexity

In this paper, a new class of higher-order (V,α,ρ,θ)-invex function is introduced. Conditions are obtained under which a fractional function is higher-order (V,α,ρ,θ)-invex. Sufficiency of Karush-Kuhn-Tucker conditions is shown under this class of function. We then consider a nondifferentiable multiobjective fractional programming problem and derive the duality theorems.     

Ultrafilters,independent collections,and applications to topology

The following are consequences of the main results in this paper:

• 1.(1) The number of countably compact, completely regular spaces of density κ is 2^22κ.
• 2.(2) There are 2^2κ points in U(κ) (= space of uniform ultrafilters on κ), each of which has tightness 2^κ in U(κ) and is a limit point of a countable subset of U(κ).
• 3.(3) There are 2^2κ points in U(κ), each of which has tightness 2^κ and is a weak P-point of κ¹.
• 4.(4) For each λ ⩽ κ there are at least 2^2λ · κ points in βκ, each of which has tightness 2^λ in β κ and is a weak P-point of κ¹. Moreover, under GCH there are at least 2^2λ · κ^λ such points.

     

Pattern formation in the Brusselator system

In the paper, we deal with a reaction-diffusion system well known as the Brusselator model and some improved results for the steady states of this model are presented. We first give an a priori estimates (positive upper and lower bounds) of positive steady states. Then, we obtain the non-existence and existence of positive non-constant steady states as the parameters λ, θ and b are varied, which means some certain conditions under which the pattern formation occurs or not.     

Simultaneous semi-eigenvectors for matrices over an extremal algebra

We consider a process withn jobs which is repeated in a periodic manner. This problem can be described by a “simultaneous semi-eigenvector problem”: Find all feasible periods λ for which there exists a time schedule x fulfilling $$\max _{u = 1}^n \left( {x_u + \alpha _{uv} } \right) \leqslant x_v $$ . Letd(λ): =x _n ?x ₁ be the minimum duration of one single process during one cycle under the restriction that the complete system is operated with period λ. We show thatd(λ) is a decreasing and piecewise linear function and we present a polynomial algorithm to calculate this function explicitly.     

On effective determination of symmetric-square lifts

Let F be the symmetric-square lift with Laplace eigenvalue λ _F (Δ) = 1+4µ². Suppose that |µ| ≤ Λ. We show that F is uniquely determined by the central values of Rankin-Selberg L-functions L(s, F ? h), where h runs over the set of holomorphic Hecke eigen cusp forms of weight κ ≡ 0 (mod 4) with κ?^?+?, t9 = max {4(1+4θ)/(1?18θ), 8(2?9θ)/3(1?18θ)} for any 0 ≤ θ < 1/18 and any ∈ > 0. Here θ is the exponent towards the Ramanujan conjecture for GL₂ Maass forms.     

Notes on the partition property of {\mathcal{P}_\kappa\lambda}

We investigate the partition property of ${\mathcal{P}_{\kappa}\lambda}$ . Main results of this paper are as follows: (1) If λ is the least cardinal greater than κ such that ${\mathcal{P}_{\kappa}\lambda}$ carries a (λ ^κ , 2)-distributive normal ideal without the partition property, then λ is ${\Pi^1_n}$ -indescribable for all n?<?ω but not ${\Pi^2_1}$ -indescribable. (2) If cf(λ) ≥?κ, then every ineffable subset of ${\mathcal{P}_{\kappa}\lambda}$ has the partition property. (3) If cf(λ) ≥ κ, then the completely ineffable ideal over ${\mathcal{P}_{\kappa}\lambda}$ has the partition property.     

Locally adaptive fitting of semiparametric models to nonstationary time series

We fit a class of semiparametric models to a nonstationary process. This class is parametrized by a mean function μ(·) and a p-dimensional function θ(·)=(θ⁽¹⁾(·),…,θ^(p)(·))′ that parametrizes the time-varying spectral density f_θ(·)(λ). Whereas the mean function is estimated by a usual kernel estimator, each component of θ(·) is estimated by a nonlinear wavelet method. According to a truncated wavelet series expansion of θ⁽ⁱ⁾(·), we define empirical versions of the corresponding wavelet coefficients by minimizing an empirical version of the Kullback–Leibler distance. In the main smoothing step, we perform nonlinear thresholding on these coefficients, which finally provides a locally adaptive estimator of θ⁽ⁱ⁾(·). This method is fully automatic and adapts to different smoothness classes. It is shown that usual rates of convergence in Besov smoothness classes are attained up to a logarithmic factor.     

Multiplicity and uniqueness of positive solutions for nonhomogeneous semilinear elliptic equation with critical exponent

In this paper, we consider the following problem

$$\left\{ {\begin{array}{*{20}{c}}{ - \Delta u\left( x \right) + u\left( x \right) = \lambda \left( {{u^p}\left( x \right) + h\left( x \right)} \right),\;x \in {\mathbb{R}^N},} \\ {u\left( x \right) \in {H^1}\left( {{\mathbb{R}^N}} \right),\;u\left( x \right) \succ 0,\;x \in {\mathbb{R}^N},\;} \end{array}} \right.\;\left( * \right)$$

, where λ > 0 is a parameter, p = (N+2)/(N?2). We will prove that there exists a positive constant 0 < λ* < +∞ such that (*) has a minimal positive solution for λ ∈ (0, λ*), no solution for λ > λ*, a unique solution for λ = λ*. Furthermore, (*) possesses at least two positive solutions when λ ∈ (0, λ*) and 3 ≤ N ≤ 5. For N ≥ 6, under some monotonicity conditions of h we show that there exists a constant 0 < λ** < λ* such that problem (*) possesses a unique solution for λ ∈ (0, λ**).

     

Gradients of inner functions in strictly pseudoconvex domains

We investigate the unbalanced ordinary partition relations of the form λ → (λ, α)² for various values of the cardinal λ and the ordinal α. For example, we show that for every infinite cardinal κ, the existence of a κ+-Suslin tree implies κ⁺ ? (κ⁺, log _κ (κ⁺) + 2)². The consistency of the positive partition relation b → (b, α)2 for all α < ω₁ for the bounding number b is also established from large cardinals.     

Indestructibility, instances of strong compactness, and level by level inequivalence

Suppose λ > κ is measurable. We show that if κ is either indestructibly supercompact or indestructibly strong, then A = {δ < κ | δ is measurable, yet δ is neither δ ⁺ strongly compact nor a limit of measurable cardinals} must be unbounded in κ. The large cardinal hypothesis on λ is necessary, as we further demonstrate by constructing via forcing two models in which ${A = \emptyset}$ . The first of these contains a supercompact cardinal κ and is such that no cardinal δ > κ is measurable, κ’s supercompactness is indestructible under κ-directed closed, (κ ⁺, ∞)-distributive forcing, and every measurable cardinal δ < κ is δ ⁺ strongly compact. The second of these contains a strong cardinal κ and is such that no cardinal δ > κ is measurable, κ’s strongness is indestructible under < κ-strategically closed, (κ ⁺, ∞)-distributive forcing, and level by level inequivalence between strong compactness and supercompactness holds. The model from the first of our forcing constructions is used to show that it is consistent, relative to a supercompact cardinal, for the least cardinal κ which is both strong and has its strongness indestructible under κ-directed closed, (κ ⁺, ∞)-distributive forcing to be the same as the least supercompact cardinal, which has its supercompactness indestructible under κ-directed closed, (κ ⁺, ∞)-distributive forcing. It further follows as a corollary of the first of our forcing constructions that it is possible to build a model containing a supercompact cardinal κ in which no cardinal δ > κ is measurable, κ is indestructibly supercompact, and every measurable cardinal δ < κ which is not a limit of measurable cardinals is δ ⁺ strongly compact.     

Estimates of the solutions of difference-differential equations of neutral type

In this paper, we study scalar difference-differential equations of neutral type of general form $$\sum\limits_{j = 0}^m {\int_0^h {u^{(j)} (t - \theta )d\sigma _j (\theta ) = 0,t > h,} } $$ where the σ_j(θ) are functions of bounded variation. For the solutions of this equation, we obtain the following estimate: $$\left\| {u(t)} \right\|W_2^m (T,T + h) \leqslant CT^{q - 1} e^{\kappa T} \left\| {u(t)} \right\|W_2^m (0,h),$$ where C is a constant independent of u ₀(t) and the values of q and ? are determined by the properties of the characteristic determinant of this equation. Earlier, this estimate was proved for equations of less general form. For example, for piecewise constant functions σ _j(θ) or for the case in which the function σ _m(θ) has jumps at both points θ = 0 and θ = h. In the present paper, this estimate is obtained under the only condition that σ _m(θ) experiences a jump at the point θ = 0; this condition is necessary for the correct solvability of the initial-value problem.     

Dichotomy for the Hausdorff dimension of the set of nonergodic directions

Given an irrational 0<λ<1, we consider billiards in the table P _λ formed by a \(\tfrac{1}{2}\times1\) rectangle with a horizontal barrier of length \(\frac{1-\lambda}{2}\) with one end touching at the midpoint of a vertical side. Let NE?(P _λ ) be the set of θ such that the flow on P _λ in direction θ is not ergodic. We show that the Hausdorff dimension of NE?(P _λ ) can only take on the values 0 and \(\tfrac{1}{2}\), depending on the summability of the series \(\sum_{k}\frac{\log\log q_{k+1}}{q_{k}}\) where {q _k } is the sequence of denominators of the continued fraction expansion of λ. More specifically, we prove that the Hausdorff dimension is \(\frac{1}{2}\) if this series converges, and 0 otherwise. This extends earlier results of Boshernitzan and Cheung.     

On uncountable cardinal sequences for superatomic Boolean algebras

The countable sequences of cardinals which arise as cardinal sequences of superatomic Boolean algebras were characterized by La Grange on the basis of ZFC set theory. However, no similar characterization is available for uncountable cardinal sequences. In this paper we prove the following two consistency results:

1. Ifθ = 〈κ _α :α <ω ₁〉 is a sequence of infinite cardinals, then there is a cardinal-preserving notion of forcing that changes cardinal exponentiation and forces the existence of a superatomic Boolean algebraB such that θ is the cardinal sequence ofB.
2. Ifκ is an uncountable cardinal such thatκ ^<κ =κ andθ = 〈κ _α :α <κ ⁺〉 is a cardinal sequence such thatκ _α ≥κ for everyα <κ ⁺ andκ _α =κ for everyα <κ ⁺ such that cf(α)<κ, then there is a cardinal-preserving notion of forcing that changes cardinal exponentiation and forces the existence of a superatomic Boolean algebraB such that θ is the cardinal sequence ofB.

     

Ranges of certain functionals in classes of regular functions

LetP _κ,n (λ,β) be the class of functions \(g(z) = 1 + \sum\nolimits_{v = n}^\infty {c_\gamma z^v }\) , regular in ¦z¦<1 and satisfying the condition $$\int_0^{2\pi } {\left| {\operatorname{Re} \left[ {e^{i\lambda } g(z) - \beta \cos \lambda } \right]} \right|} /\left( {1 - \beta } \right)\cos \lambda \left| {d\theta \leqslant \kappa \pi ,} \right.z = re^{i\theta } ,$$ , 0 < r < 1 (κ?2,n?1, 0?Β<1, -π<λ<π/2;M _κ,n (λ,β,α),n?2, is the class of functions \(f(z) = z + \sum\nolimits_{v = n}^\infty {a_v z^v }\) , regular in¦z¦<1 and such thatF _α(z)∈P _κ,n?1(λ,β), where \(F_\alpha (z) = (1 - \alpha )\frac{{zf'(z)}}{{f(z)}} + \alpha (1 + \frac{{zf'(z)}}{{f'(z)}})\) (0?α?1). Onr considers the problem regarding the range of the system {g ^(v?1)(z_?)/(v?1)!}, ?=1,2,...,m,v=1,2,...,N _?, on the classP _κ,1(λ,β). On the classesP _κ,n (λ,β),M _κ,n (λ,β,α) one finds the ranges of C_v, v?n, a_m, n?m?2n-2, and ofg(?),F _?(?), 0<¦ξ¦<1, ξ is fixed.     

A note on singular nonlinear boundary value problems

In this paper we study the existence and multiplicity of solutions of the following operator equation in Banach space E:

u=λAu,0<λ<+∞,u∈P?{θ},     

Extinction times for a birth–death process with weak competition

We consider a birth–death process with birth rates iλ and death rates iμ+i(i?1)θ, where i is the current state of the process. A positive competition rate θ is assumed to be small. In the supercritical case where λ > μ, this process can be viewed as a demographic model for a population with high carrying capacity around (λ?μ)/θ. The article reports in a self-contained manner on the asymptotic properties of the time to extinction for this logistic branching process as θ → 0. All three reproduction regimes λ > μ, λ < μ, and λ = μ are studied.     

Strong tree properties for two successive cardinals

An inaccessible cardinal κ is supercompact when (κ, λ)-ITP holds for all λ?≥ κ. We prove that if there is a model of ZFC with two supercompact cardinals, then there is a model of ZFC where simultaneously ${(\aleph_2, \mu)}$ -ITP and ${(\aleph_3, \mu')}$ -ITP hold, for all ${\mu\geq \aleph_2}$ and ${\mu'\geq \aleph_3}$ .