Indestructibility and destructible measurable cardinals

Say that \({\kappa}\)’s measurability is destructible if there exists a < \({\kappa}\)-closed forcing adding a new subset of \({\kappa}\) which destroys \({\kappa}\)’s measurability. For any δ, let λ_δ =_df The least beth fixed point above δ. Suppose that \({\kappa}\) is indestructibly supercompact and there is a measurable cardinal λ > \({\kappa}\). It then follows that \({A_{1} = \{\delta < \kappa \mid \delta}\) is measurable, δ is not a limit of measurable cardinals, δ is not δ⁺ strongly compact, and δ’s measurability is destructible when forcing with partial orderings having rank below λ_δ} is unbounded in \({\kappa}\). On the other hand, under the same hypotheses, \({A_{2} = \{\delta < \kappa \mid \delta}\) is measurable, δ is not a limit of measurable cardinals, δ is not δ⁺ strongly compact, and δ′s measurability is indestructible when forcing with either Add(δ, 1) or Add(δ, δ⁺)} is unbounded in \({\kappa}\) as well. The large cardinal hypothesis on λ is necessary, as we further demonstrate by constructing via forcing two distinct models in which either \({A_{1} = \emptyset}\) or \({A_{2} = \emptyset}\). In each of these models, both of which have restricted large cardinal structures above \({\kappa}\), every measurable cardinal δ which is not a limit of measurable cardinals is δ⁺ strongly compact, and there is an indestructibly supercompact cardinal \({\kappa}\). In the model in which \({A_{1} = \emptyset}\), every measurable cardinal δ which is not a limit of measurable cardinals is <λ_δ strongly compact and has its <λ_δ strong compactness (and hence also its measurability) indestructible when forcing with δ-directed closed partial orderings having rank below λ_δ. The choice of the least beth fixed point above δ is arbitrary, and other values of λ_δ are also possible.     

Martin’s maximum revisited

We present several results relating the general theory of the stationary tower forcing developed by Woodin with forcing axioms. In particular we show that, in combination with class many Woodin cardinals, the forcing axiom MM⁺⁺ makes the \({\Pi_2}\)-fragment of the theory of \({H_{\aleph_2}}\) invariant with respect to stationary set preserving forcings that preserve BMM. We argue that this is a promising generalization to \({H_{\aleph_2}}\) of Woodin’s absoluteness results for \({L(\mathbb{R})}\). In due course of proving this, we shall give a new proof of some of these results of Woodin. Finally we relate our generic absoluteness results with the resurrection axioms introduced by Hamkins and Johnstone and with their unbounded versions introduced by Tsaprounis.     

Kleisli Monoids Describing Approach Spaces

We study the functional ideal monad \(\mathbb {I} = (\mathsf {I}, m, e)\) on S e t and show that this monad is power-enriched. This leads us to the category \(\mathbb {I}\)- M o n of all \(\mathbb {I}\)-monoids with structure preserving maps. We show that this category is isomorphic to A p p, the category of approach spaces with contractions as morphisms. Through the concrete isomorphism, an \(\mathbb {I}\)-monoid (X,ν) corresponds to an approach space \((X, \mathfrak {A}),\) described in terms of its bounded local approach system. When I is extended to R e l using the Kleisli extension \(\check {\mathsf {I}},\) from the fact that \(\mathbb {I}\)- M o n and \((\mathbb {I},2)\)- C a t are isomorphic, we obtain the result that A p p can be isomorphically described in terms of convergence of functional ideals, based on the two axioms of relational algebras, reflexivity and transitivity. We compare these axioms to the ones put forward in Lowen (2015). Considering the submonad \(\mathbb {B}\) of all prime functional ideals, we show that it is both sup-dense and interpolating in \(\mathbb {I}\), from which we get that \((\mathbb {I},2)\)- C a t and \((\mathbb {B},2)\)- C a t are isomorphic. We present some simple axioms describing A p p in terms of prime functional ideal convergence.     

Weak square and stationary reflection

It is well-known that the square principle \({\square_\lambda}\) entails the existence of a non-reflecting stationary subset of λ⁺, whereas the weak square principle \({\square^{*} _\lambda}\) does not. Here we show that if μ^cf(λ) < λ for all μ < λ, then \({\square^{*} _\lambda}\) entails the existence of a non-reflecting stationary subset of \({E^{\lambda^+}_{{\rm cf}(\lambda)}}\)in the forcing extension for adding a single Cohen subset of λ⁺.It follows that indestructible forms of simultaneous stationary reflection entail the failure of weak square. We demonstrate this by settling a question concerning the subcomplete forcing axiom (SCFA), proving that SCFA entails the failure of \({\square^{*} _\lambda}\) for every singular cardinal λ of countable cofinality.     

Relative Projectivity and Transferability for Partial Lattices

A partial lattice P is ideal-projective, with respect to a class \(\mathcal {C}\) of lattices, if for every \(K\in \mathcal {C}\) and every homomorphism φ of partial lattices from P to the ideal lattice of K, there are arbitrarily large choice functions f:P→K for φ that are also homomorphisms of partial lattices. This extends the traditional concept of (sharp) transferability of a lattice with respect to \(\mathcal {C}\). We prove the following: (1) A finite lattice P, belonging to a variety \(\mathcal {V}\), is sharply transferable with respect to \(\mathcal {V}\) iff it is projective with respect to \(\mathcal {V}\) and weakly distributive lattice homomorphisms, iff it is ideal-projective with respect to \(\mathcal {V}\), (2) Every finite distributive lattice is sharply transferable with respect to the class \(\mathcal {R}_{\text {mod}}\) of all relatively complemented modular lattices, (3) The gluing D ₄ of two squares, the top of one being identified with the bottom of the other one, is sharply transferable with respect to a variety \(\mathcal {V}\) iff \(\mathcal {V}\) is contained in the variety \(\mathcal {M}_{\omega }\) generated by all lattices of length 2, (4) D ₄ is projective, but not ideal-projective, with respect to \(\mathcal {R}_{\text {mod}}\) , (5) D ₄ is transferable, but not sharply transferable, with respect to the variety \(\mathcal {M}\) of all modular lattices. This solves a 1978 problem of G. Grätzer, (6) We construct a modular lattice whose canonical embedding into its ideal lattice is not pure. This solves a 1974 problem of E. Nelson.     

Nonlinear *-Lie-type derivations on standard operator algebras

Let \(\mathcal{H}\) be an infinite dimensional complex Hilbert space and \(\mathcal{A}\) be a standard operator algebra on \(\mathcal{H}\) which is closed under the adjoint operation. It is shown that each nonlinear *-Lie-type derivation δ on \(\mathcal{A}\) is a linear *-derivation. Moreover, δ is an inner *-derivation as well.     

Lie Triple Derivations on von Neumann Algebras

Let A be a von Neumann algebra with no central abelian projections. It is proved that if an additive map δ :A → A satisfies δ([[a, b], c]) = [[δ(a), b], c] + [[a, δ(b)], c] +[[a, b], δ(c)] for any a, b, c∈ A with ab = 0(resp. ab = P, where P is a fixed nontrivial projection in A), then there exist an additive derivation d from A into itself and an additive map f :A → ZA vanishing at every second commutator [[a, b], c] with ab = 0(resp.ab = P) such that δ(a) = d(a) + f(a) for any a∈ A.     

Smoothness of solutions of almost hypoelliptic equations

The paper studies a class of almost hypoelliptic equations P(D)U = ? in a strip. It is proved that for \(\mathcal{H}\) great enough and for δ > 0 small enough all solutions of this equation, which are square summable with the weight e ^?δ|x| and for which \(D_2^{\alpha _2 } U\), where α ₂ = 0, …, \(ord_{\alpha _2 } P\), are infinitely differentiable in x ₁ functions, provided D ₁ ^j ? ∈ L ₂(\(\Omega _\mathcal{H} \)) for any j.     

Sets with even partition functions and 2-adic integers

For P ? \(\mathbb{F}_2 \)[z] with _P(0) = 1 and deg(_P) ≥ 1, let \(\mathcal{A}\) = \(\mathcal{A}\)(P) (cf. [4], [5], [13]) be the unique subset of ? such that Σ _n≥0 p(\(\mathcal{A}\), n)z ⁿ ≡ P(z) (mod 2), where p(\(\mathcal{A}\), n) is the number of partitions of n with parts in \(\mathcal{A}\). Let p be an odd prime and P ? \(\mathbb{F}_2 \)[z] be some irreducible polynomial of order p, i.e., p is the smallest positive integer such that P(z) divides 1 + z ^p in \(\mathbb{F}_2 \)[z]. In this paper, we prove that if m is an odd positive integer, the elements of \(\mathcal{A}\) = \(\mathcal{A}\)(P) of the form 2 ^k m are determined by the 2-adic expansion of some root of a polynomial with integer coefficients. This extends a result of F. Ben Saïd and J.-L. Nicolas [6] to all primes p.     

Elliptic ovoids and their rosettes in a classical generalized quadrangle of even order

Let Q₀ be the classical generalized quadrangle of order q = 2ⁿ(n≥2) arising from a non-degenerate quadratic form in a 5-dimensional vector space defined over a finite field of order q. We consider the rank two geometry \(\mathcal {X}\) having as points all the elliptic ovoids of Q₀ and as lines the maximal pencils of elliptic ovoids of Q₀ pairwise tangent at the same point. We first prove that \(\mathcal {X}\) is isomorphic to a 2-fold quotient of the affine generalized quadrangle Q?Q₀, where Q is the classical (q,q²)-generalized quadrangle admitting Q₀ as a hyperplane. Further, we classify the cliques in the collinearity graph Γ of \(\mathcal {X}\). We prove that any maximal clique in Γ is either a line of \(\mathcal {X}\) or it consists of 6 or 4 points of \(\mathcal {X}\) not contained in any line of \(\mathcal {X}\), accordingly as n is odd or even. We count the number of cliques of each type and show that those cliques which are not contained in lines of \(\mathcal {X}\) arise as subgeometries of Q defined over \(\mathbb {F}_{2}\).     

On Causal Invertibility with Respect to a Cone of Integral-Difference Operators in Vector Function Spaces

Let \(\mathbb{S}\) be a cone in ? ⁿ . A bounded linear operator T: L _p (? ⁿ ) → L _p (? ⁿ ) is said to be causal with respect to \(\mathbb{S}\) if the implication x(s) = 0 (s ε W ? \(\mathbb{S}\)) ? (Tx) (s) = 0 (s ε W ? \(\mathbb{S}\)) is valid for any x ε L _p (? ⁿ ) and any open subset W\(\subseteq\) ? ⁿ . The set of all causal operators is a Banach algebra. We describe the spectrum of the operator

$(Tx)(t) = \sum\limits_{n = 1}^\infty {a_n x(t - t_n )} + \int {\mathbb{S}g(s)x(t - s)ds,} \quad t \in \mathbb{R}^n ,$

in this algebra. Here x ranges in a Banach space \(\mathbb{E}\), the a _n are bounded linear operators in \(\mathbb{E}\), and the function g ranges in the set of bounded operators in \(\mathbb{E}\).

     

Integral means of weighted Hardy–Bloch functions

Let \({\mathcal{B}^\omega(p, q, B_d)}\) denote the \({\omega}\)-weighted Hardy–Bloch space on the unit ball B _d of \({\mathbb{C}^d}\), \({d\ge 1}\). For \({2< p,q < \infty}\) and \({f\in \mathcal{B}^\omega(p, q, B_d)}\), we obtain sharp estimates on the growth of the p-integral means M _p (f, r) as \({r\to 1-}\).     

On a two-phase free boundary condition for <Emphasis Type="Italic">p</Emphasis>-harmonic measures

Let \({\Omega^i\subset {\bf R}^n, i\in\{1,2\}}\) , be two (δ, r ₀)-Reifenberg flat domains, for some \({0 < \delta < \hat \delta}\) and r ₀ > 0, assume \({\Omega^1\cap\Omega^2=\emptyset}\) and that, for some \({w\in {\bf R}^n}\) and some 0 < r, \({w\in\partial\Omega^1\cap\partial\Omega^2, \partial\Omega^1\cap B(w,2r)=\partial\Omega^2\cap B(w,2r)}\) . Let p, 1 < p < ∞, be given and let u ⁱ , \({i\in\{1,2\}}\) , denote a non-negative p-harmonic function in Ω ⁱ , assume that u ⁱ , \({i\in\{1,2\}}\), is continuous in \({\bar\Omega^i\cap B(w,2r) }\) and that u ⁱ  = 0 on \({\partial\Omega^i\cap B(w,2r)}\) . Extend u ⁱ to B(w, 2r) by defining \({u^i\equiv 0}\) on \({B(w,2r) {\setminus} \Omega^i}\). Then there exists a unique finite positive Borel measure μ ⁱ , \({i\in\{1,2\}}\) , on R ⁿ , with support in \({\partial\Omega^i\cap B(w,2r)}\) , such that if \({\phi \in C_0^\infty (B(w,2r))}\) , then

$\int\limits_{\mathbf R^n} \,|\nabla u^i|^{ p-2} \,\langle \nabla u^i, \,\nabla \phi \rangle \,dx =- \int\limits_{\mathbf R^n} \,\phi \,d \mu^i.$

Let \({\Delta(w,2r)=\partial\Omega^1\cap B(w,2r)=\partial\Omega^2\cap B(w,2r)}\) . The main result proved in this paper is the following. Assume that μ ² is absolutely continuous with respect to μ ¹ on Δ(w, 2r), d μ ² = kd μ ¹ for μ ¹-almost every point in Δ(w, 2r) and that \({\log k\in VMO(\Delta(w,r),\mu^1)}\) . Then there exists \({\tilde \delta = \tilde \delta(p,n) > 0}\) , \({\tilde \delta < \hat \delta}\) , such that if \({\delta\leq\tilde\delta}\) , then Δ(w, r/2) is Reifenberg flat with vanishing constant. Moreover, the special case p = 2, i.e., the linear case and the corresponding problem for harmonic measures, has previously been studied in Kenig and Toro (J Reine Angew Math 596:1–44, 2006).

     

Ordinal indices of Small Subspaces of <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>

We calculate the ordinal L _p index defined in [3] for Rosenthal’s space X _p , \({\ell_p}\) and \({\ell_2}\). We show that an infinite-dimensional subspace of L _p \({(2 < p < \infty)}\) non-isomorphic to \({\ell_2}\) embeds in \({\ell_p}\) if and only if its ordinal index is the minimal possible. We also give a sufficient condition for a \({\mathcal{L}_p}\) subspace of \({\ell_p \oplus \ell_2}\) to be isomorphic to X _p .     

On a theorem of Rigby

Let \({\Delta = BAG(2, q)}\) denote the classical biaffine plane of order q, that is, the symmetric \({((q^2 - 1)_q)}\) configuration obtained from the classical affine plane \({\Sigma = AG(2, q)}\) of order q by omitting a point of \({\Sigma}\) together with all lines through this point. Now let \({q \geq 4}\) be a power of a prime p and assume that \({\Delta}\) admits an embedding into the projective plane \({\Pi = PG(2, F)}\), where F is a (not necessarily commutative) field. Then this embedding extends to a projective subplane \({\Pi_0 \cong PG(2, q)}\) of \({\Pi}\); in particular, F has characteristic p. Consequently, \({BAG(2, q)}\) with \({q\geq 4}\) admits an embedding into \({PG(2, q')}\) if only if q′ is a power of q. This strengthens a result of Rigby (Canad J Math 17:977–1009, 1965) in a special case while simultaneously providing a more elegant proof.     

The Minimal Size of Infinite Maximal Antichains in Direct Products of Partial Orders

For a partial order \(\mathbb {P}\) having infinite antichains by \(\mathfrak {a}(\mathbb {P})\) we denote the minimal cardinality of an infinite maximal antichain in \(\mathbb {P}\) and investigate how does this cardinal invariant of posets behave in finite products. In particular we show that \(\min \{ \mathfrak {a}(\mathbb {P}),\mathfrak {p} (\text {sq} \mathbb {P}) \} \leq \mathfrak {a} (\mathbb {P}^{n} ) \leq \mathfrak {a} (\mathbb {P})\), for all \(n\in \mathbb {N}\), where \(\mathfrak {p} (\text {sq} \mathbb {P})\) is the minimal size of a centered family without a lower bound in the separative quotient of the poset \(\mathbb {P}\), or \(\mathfrak {p} (\text {sq} \mathbb {P})=\infty \), if there is no such family. So we have \(\mathfrak {a} (\mathbb {P} \times \mathbb {P})=\mathfrak {a} (\mathbb {P})\) whenever \(\mathfrak {p} (\text {sq} \mathbb {P})\geq \mathfrak {a} (\mathbb {P})\) and we show that, in addition, this equality holds for all posets obtained from infinite Boolean algebras of size ≤ø ₁ by removing zero, all reversed trees, all atomic posets and, in particular, for all posets of the form \(\langle \mathcal {C} ,\subset \rangle \), where \(\mathcal {C}\) is a family of nonempty closed sets in a compact T ₁-space containing all singletons. As a by-product we obtain the following combinatorial statement: If X is an infinite set and {A _i ×B _i :i∈I} an infinite partition of the square X ², then at least one of the families {A _i :i∈I} and {B _i :i∈I} contains an infinite partition of X.     

Idempotent Ideals and the Igusa-Todorov Functions

Let Λ be an artin algebra and \(\mathfrak {A}\) a two-sided idempotent ideal of Λ, that is, \(\mathfrak {A}\) is the trace of a projective Λ-module P in Λ. We consider the categories of finitely generated modules over the associated rings \({\Lambda }/\mathfrak {A}, {\Lambda }\) and Γ = End_Λ(P) ^{o p} and study the relationship between their homological properties via the Igusa-Todorov functions.     

Holomorphic invariant forms of a bounded domain

Given a complete ortho-normal system  = (0, 1, 2, . . .) of L2H(D), the space of holomorphic and absolutely square integrable functions in the bounded domain D of Cn, we construct a holomorphic imbedding ι : D →■(n, ∞), the complex infinite dimensional Grassmann manifold of rank n. It is known that in ■(n, ∞) there are n closed (μ, μ)-forms τμ (μ = 1, . . . , n) which are invariant under the holomorphic isometric automorphism of ■(n, ∞) and generate algebraically all the harmonic differential forms of ...     

On estimation of delay location

Assume that we observe a stationary Gaussian process X(t), \({t \in [-r, T]}\) , which satisfies the affine stochastic delay differential equation

$d X(t) = \int\limits_{[-r,0]}X(t+u)\, a_\vartheta (du)\,dt +dW(t), \quad t\ge 0,$

where W(t), t ≥ 0, is a standard Wiener process independent of X(t), \({t\in [-r, 0]}\) , and \({a_\vartheta}\) is a finite signed measure on [?r, 0], \({\vartheta\in\Theta}\) . The parameter \({\vartheta}\) is unknown and has to be estimated based on the observation. In this paper we consider the case where \({\Theta=(\vartheta_0,\vartheta_1)}\) , \({-\infty\,<\,\vartheta_0 <0 \,<\,\vartheta_1\,<\,\infty}\) , and the measures \({a_\vartheta}\) are of the form

$a_\vartheta = a+b_\vartheta-b,$

where a and b are finite signed measure on [?r, 0] and \({b_\vartheta}\) is the translate of b by \({\vartheta}\) . We study the limit behaviour of the normalized likelihoods

$Z_{T,\vartheta}(u) = \frac{dP_T^{\vartheta+\delta_T u}}{dP_T^\vartheta}$

as T→ ∞, where \({P_T^\vartheta}\) is the distribution of the observation if the true value of the parameter is \({\vartheta}\) . A necessary and sufficient condition for the existence of a rescaling function δ _T such that \({Z_{T,\vartheta}(u)}\) converges in distribution to an appropriate nondegenerate limiting function \({Z_{\vartheta}(u)}\) is found. It turns out that then the limiting function \({Z_{\vartheta}(u)}\) is of the form

$Z_\vartheta(u)=\exp\left(B^H(u) - E[B^H(u)]^2/2\right),$

where \({H\in[1/2,1]}\) and B ^H (u), \({u\in\mathbb{R}}\) , is a fractional Brownian motion with index H, and δ _T  = T ^?1/(2H) ?(T) with a slowly varying function ?. Every \({H\in[1/2,1]}\) may occur in this framework. As a consequence, the asymptotic behaviour of maximum likelihood and Bayes estimators is found.

     

Rational approximation to surfaces defined by polynomials in one variable

For a Tychonoff space X, we denote by C _p (X) the space of all real-valued continuous functions on X with the topology of pointwise convergence.

In this paper we prove that:

• If every finite power of X is Lindelöf then C _p (X) is strongly sequentially separable iff X is \({\gamma}\)-set.
• \({B_{\alpha}(X)}\) (= functions of Baire class \({\alpha}\) (\({1 < \alpha \leq \omega_1}\)) on a Tychonoff space X with the pointwise topology) is sequentially separable iff there exists a Baire isomorphism class \({\alpha}\) from a space X onto a \({\sigma}\)-set.
• \({B_{\alpha}(X)}\) is strongly sequentially separable iff \({iw(X)=\aleph_0}\) and X is a \({Z^{\alpha}}\)-cover \({\gamma}\)-set for \({0 < \alpha \leq \omega_1}\).
• There is a consistent example of a set of reals X such that C _p (X) is strongly sequentially separable but B₁(X) is not strongly sequentially separable.
• B(X) is sequentially separable but is not strongly sequentially separable for a \({\mathfrak{b}}\)-Sierpiński set X.