Antibasis theorems for {\Pi^0_1} classes and the jump hierarchy

We prove two antibasis theorems for ${\Pi^0_1}$ classes. The first is a jump inversion theorem for ${\Pi^0_1}$ classes with respect to the global structure of the Turing degrees. For any ${P\subseteq 2^\omega}$ , define S(P), the degree spectrum of P, to be the set of all Turing degrees a such that there exists ${A \in P}$ of degree a. For any degree ${{\bf a \geq 0'}}$ , let ${\textrm{Jump}^{-1}({\bf a) = \{b : b' = a \}}}$ . We prove that, for any ${{\bf a \geq 0'}}$ and any ${\Pi^0_1}$ class P, if ${\textrm{Jump}^{-1} ({\bf a}) \subseteq S(P)}$ then P contains a member of every degree. For any degree ${{\bf a \geq 0'}}$ such that a is recursively enumerable (r.e.) in 0', let ${Jump_{\bf \leq 0'} ^{-1}({\bf a)=\{b : b \leq 0' \textrm{and} b' = a \}}}$ . The second theorem concerns the degrees below 0'. We prove that for any ${{\bf a\geq 0'}}$ which is recursively enumerable in 0' and any ${\Pi^0_1}$ class P, if ${\textrm{Jump}_{\bf \leq 0'} ^{-1}({\bf a)} \subseteq S(P)}$ then P contains a member of every degree.     

Howe correspondence and Springer correspondence for real reductive dual pairs

We consider a real reductive dual pair (G′, G) of type I, with rank ${({\rm G}^{\prime}) \leq {\rm rank(G)}}$ . Given a nilpotent coadjoint orbit ${\mathcal{O}^{\prime} \subseteq \mathfrak{g}^{{\prime}{*}}}$ , let ${\mathcal{O}^{\prime}_\mathbb{C} \subseteq \mathfrak{g}^{{\prime}{*}}_\mathbb{C}}$ denote the complex orbit containing ${\mathcal{O}^{\prime}}$ . Under some condition on the partition λ′ parametrizing ${\mathcal{O}^{\prime}}$ , we prove that, if λ is the partition obtained from λ by adding a column on the very left, and ${\mathcal{O}}$ is the nilpotent coadjoint orbit parametrized by λ, then ${\mathcal{O}_\mathbb{C}= \tau (\tau^{\prime -1}(\mathcal{O}_\mathbb{C}^{\prime}))}$ , where ${\tau, \tau^{\prime}}$ are the moment maps. Moreover, if ${chc(\hat\mu_{\mathcal{O}^{\prime}}) \neq 0}$ , where chc is the infinitesimal version of the Cauchy-Harish-Chandra integral, then the Weyl group representation attached by Wallach to ${\mu_{\mathcal{O}^{\prime}}}$ with corresponds to ${\mathcal{O}_\mathbb{C}}$ via the Springer correspondence.     

Weak partition properties on trees

We investigate the following weak Ramsey property of a cardinal κ: If χ is coloring of nodes of the tree κ ^<ω by countably many colors, call a tree ${T \subseteq \kappa^{ < \omega}}$ χ-homogeneous if the number of colors on each level of T is finite. Write ${\kappa \rightsquigarrow (\lambda)^{ < \omega}_{\omega}}$ to denote that for any such coloring there is a χ-homogeneous λ-branching tree of height ω. We prove, e.g., that if ${\kappa < \mathfrak{p}}$ or ${\kappa > \mathfrak{d}}$ is regular, then ${{\kappa \rightsquigarrow (\kappa)^{ < \omega}_{\omega}}}$ and that ${\mathfrak{b}}$ ${(\mathfrak{b})^{ < \omega}_{\omega}}$ and ${\mathfrak{d}}$ ${(\mathfrak{d})^{ < \omega}_{\omega}}$ . The arrow is applied to prove a generalization of a theorem of Hurewicz: A ?ech-analytic space is σ-locally compact iff it does not contain a closed homeomorphic copy of irrationals.     

On the boundary of the attainable set of the dirichlet spectrum

Denoting by ${\varepsilon\subseteq\mathbb{R}^2}$ the set of the pairs ${(\lambda_1(\Omega),\,\lambda_2(\Omega))}$ for all the open sets ${\Omega\subseteq\mathbb{R}^N}$ with unit measure, and by ${\Theta\subseteq\mathbb{R}^N}$ the union of two disjoint balls of half measure, we give an elementary proof of the fact that ${\partial\varepsilon}$ has horizontal tangent at its lowest point ${(\lambda_1(\Theta),\,\lambda_2(\Theta))}$ .     

A covering lemma for {K(\mathbb {R})}

The Dodd–Jensen Covering Lemma states that “if there is no inner model with a measurable cardinal, then for any uncountable set of ordinals X, there is a ${Y\in K}$ such that ${X\subseteq Y}$ and |X| = |Y|”. Assuming ZF+AD alone, we establish the following analog: If there is no inner model with an ${\mathbb {R}}$ –complete measurable cardinal, then the real core model ${K(\mathbb {R})}$ is a “very good approximation” to the universe of sets V; that is, ${K(\mathbb {R})}$ and V have exactly the same sets of reals and for any set of ordinals X with ${|{X}|\ge\Theta}$ , there is a ${Y\in K(\mathbb {R})}$ such that ${X\subseteq Y}$ and |X| = |Y|. Here ${\mathbb {R}}$ is the set of reals and ${\Theta}$ is the supremum of the ordinals which are the surjective image of ${\mathbb {R}}$ .     

Some results on metric n-Lie algebras

We study the structure of a metric n-Lie algebra G over the complex field C. Let G = SR be the Levi decomposition, where R is the radical of G and S is a strong semisimple subalgebra of G. Denote by m(G) the number of all minimal ideals of an indecomposable metric n-Lie algebra and R ⊥ the orthogonal complement of R. We obtain the following results. As S-modules, R ⊥ is isomorphic to the dual module of G/R. The dimension of the vector space spanned by all nondegenerate invariant symmetric bilinear forms on G is equal to that of the vector space of certain linear transformations on G; this dimension is greater than or equal to m(G) + 1. The centralizer of R in G is equal to the sum of all minimal ideals; it is the direct sum of R ⊥ and the center of G. Finally, G has no strong semisimple ideals if and only if R⊥■R.     

Maximal, potential and singular operators in vanishing generalized Morrey spaces

We introduce vanishing generalized Morrey spaces ${V\mathcal{L}^{p,\varphi}_\Pi (\Omega), \Omega \subseteq \mathbb{R}^n}$ with a general function ${\varphi(x, r)}$ defining the Morrey-type norm. Here ${\Pi \subseteq \Omega}$ is an arbitrary subset in Ω including the extremal cases ${\Pi = \{x_0\}, x_0 \in \Omega}$ and Π = Ω, which allows to unify vanishing local and global Morrey spaces. In the spaces ${V\mathcal{L}^{p,\varphi}_\Pi (\mathbb{R}^n)}$ we prove the boundedness of a class of sublinear singular operators, which includes Hardy-Littlewood maximal operator and Calderon-Zygmund singular operators with standard kernel. We also prove a Sobolev-Spanne type ${V\mathcal{L}^{p,\varphi}_\Pi (\mathbb{R}^n) \rightarrow V\mathcal{L}^{q,\varphi^\frac{q}{p}}_\Pi (\mathbb{R}^n)}$ -theorem for the potential operator I ^α . The conditions for the boundedness are given in terms of Zygmund-type integral inequalities on ${\varphi(x, r)}$ . No monotonicity type condition is imposed on ${\varphi(x, r)}$ . In case ${\varphi}$ has quasi- monotone properties, as a consequence of the main results, the conditions of the boundedness are also given in terms of the Matuszeska-Orlicz indices of the function ${\varphi}$ . The proofs are based on pointwise estimates of the modulars defining the vanishing spaces     

Stability of contact discontinuity for steady Euler system in infinite duct

In this paper, we prove stability of contact discontinuities for full Euler system. We fix a flat duct ${\mathcal{N}_0}$ of infinite length in ${\mathbb{R}^2}$ with width W ₀ and consider two uniform subsonic flow ${{U_l}^{\pm}=(u_l^{\pm}, 0, pl,\rho_l^{\pm})}$ with different horizontal velocity in ${\mathcal{N}_0}$ divided by a flat contact discontinuity ${\Gamma_{cd}}$ . And, we slightly perturb the boundary of ${\mathcal{N}_0}$ so that the width of the perturbed duct converges to ${W_0+\omega}$ for ${|\omega| < \delta}$ at ${x=\infty}$ for some ${\delta >0 }$ . Then, we prove that if the asymptotic state at left far field is given by ${{U_l}^{\pm}}$ , and if the perturbation of boundary of ${\mathcal{N}_0}$ and ${\delta}$ is sufficiently small, then there exists unique asymptotic state ${{U_r}^{\pm}}$ with a flat contact discontinuity ${\Gamma_{cd}^*}$ at right far field( ${x=\infty}$ ) and unique weak solution ${U}$ of the Euler system so that U consists of two subsonic flow with a contact discontinuity in between, and that U converges to ${{U_l}^{\pm}}$ and ${{U_r}^{\pm}}$ at ${x=-\infty}$ and ${x=\infty}$ respectively. For that purpose, we establish piecewise C ¹ estimate across a contact discontinuity of a weak solution to Euler system depending on the perturbation of ${\partial\mathcal{N}_0}$ and ${\delta}$ .     

Some 0/1 polytopes need exponential size extended formulations

We prove that there are 0/1 polytopes ${P \subseteq \mathbb{R}^{n}}$ that do not admit a compact LP formulation. More precisely we show that for every n there is a set ${X \subseteq \{ 0,1\}^n}$ such that conv(X) must have extension complexity at least ${2^{n/2\cdot(1-o(1))}}$ . In other words, every polyhedron Q that can be linearly projected on conv(X) must have exponentially many facets. In fact, the same result also applies if conv(X) is restricted to be a matroid polytope. Conditioning on ${\mathbf{NP}\not\subseteq \mathbf{P_{/poly}}}$ , our result rules out the existence of a compact formulation for any ${\mathbf{NP}}$ -hard optimization problem even if the formulation may contain arbitrary real numbers.     

Nikodym Maximal Functions Associated with Variable Planes in {\mathbb{R}^{3}}

Given a vector field ${\mathfrak{a}}$ on ${\mathbb{R}^3}$ , we consider a mapping ${x\mapsto \Pi_{\mathfrak{a}}(x)}$ that assigns to each ${x\in\mathbb{R}^3}$ , a plane ${\Pi_{\mathfrak{a}}(x)}$ containing x, whose normal vector is ${\mathfrak{a}(x)}$ . Associated with this mapping, we define a maximal operator ${\mathcal{M}^{\mathfrak{a}}_N}$ on ${L^1_{loc}(\mathbb{R}^3)}$ for each ${N\gg 1}$ by $$\mathcal{M}^{\mathfrak{a}}_Nf(x)=\sup_{x\in\tau} \frac{1}{|\tau|} \int_{\tau}|f(y)|\,dy$$ where the supremum is taken over all 1/N ×? 1/N?× 1 tubes τ whose axis is embedded in the plane ${\Pi_\mathfrak{a}(x)}$ . We study the behavior of ${\mathcal{M}^{\mathfrak{a}}_N}$ according to various vector fields ${\mathfrak{a}}$ . In particular, we classify the operator norms of ${\mathcal{M}^{\mathfrak{a}}_N}$ on ${L^2(\mathbb{R}^3)}$ when ${\mathfrak{a}(x)}$ is the linear function of the form (a ₁₁ x ₁?+?a ₂₁ x ₂, a ₁₂ x ₁?+?a ₂₂ x ₂, 1). The operator norm of ${\mathcal{M}^\mathfrak{a}_N}$ on ${L^2(\mathbb{R}^3)}$ is related with the number given by $$D=(a_{12}+a_{21})^2-4a_{11}a_{22}.$$      

\alpha -Admissibility for Ritt Operators

Let $T:X\rightarrow X$ be a power bounded operator on Banach space. An operator $C:X\rightarrow Y$ is called admissible for $T$ if it satisfies an estimate $\sum _k\Vert CT^k(x)\Vert ^2\,\le M^2\Vert x\Vert ^2$ . Following Harper and Wynn, we study the validity of a certain Weiss conjecture in this discrete setting. We show that when $X$ is reflexive and $T$ is a Ritt operator satisfying a appropriate square function estimate, $C$ is admissible for $T$ if and only if it satisfies a uniform estimate $(1-\vert \omega \vert ^2)^{\frac{1}{2}}\Vert C(I-\omega T)^{-1}\Vert \,\le K\,$ for $\omega \in \mathbb{C }$ , $\vert \omega \vert <1$ . We extend this result to the more general setting of $\alpha $ -admissibility. Then we investigate a natural variant of admissibility involving $R$ -boundedness and provide examples to which our general results apply.     

Morrey-type regularity of solutions to parabolic problems with discontinuous data

We consider regular oblique derivative problem in cylinder Q _T ?=????× (0, T), ${\Omega\subset {\mathbb R}^n}$ for uniformly parabolic operator ${{{\mathfrak P}}=D_t- \sum_{i,j=1}^n a^{ij}(x)D_{ij}}$ with VMO principal coefficients. Its unique strong solvability is proved in Manuscr. Math. 203?C220 (2000), when ${{{\mathfrak P}}u\in L^p(Q_T)}$ , ${p\in(1,\infty)}$ . Our aim is to show that the solution belongs to the generalized Sobolev?CMorrey space ${W^{2,1}_{p,\omega}(Q_T)}$ , when ${{{\mathfrak P}}u\in L^{p,\omega} (Q_T)}$ , ${p\in (1, \infty)}$ , ${\omega(x,r):\,{\mathbb R}^{n+1}_+\to {\mathbb R}_+}$ . For this goal an a priori estimate is obtained relying on explicit representation formula for the solution. Analogous result holds also for the Cauchy?CDirichlet problem.     

Infinite-dimensional quasigroups of finite orders

Let Σ be a finite set of cardinality k > 0, let $\mathbb{A}$ be a finite or infinite set of indices, and let $\mathcal{F} \subseteq \Sigma ^\mathbb{A}$ be a subset consisting of finitely supported families. A function $f:\Sigma ^\mathbb{A} \to \Sigma$ is referred to as an $\mathbb{A}$ -quasigroup (if $\left| \mathbb{A} \right| = n$ , then an n-ary quasigroup) of order k if $f\left( {\bar y} \right) \ne f\left( {\bar z} \right)$ for any ordered families $\bar y$ and $\bar z$ that differ at exactly one position. It is proved that an $\mathbb{A}$ -quasigroup f of order 4 is reducible (representable as a superposition) or semilinear on every coset of $\mathcal{F}$ . It is shown that the quasigroups defined on Σ^?, where ? are positive integers, generate Lebesgue nonmeasurable subsets of the interval [0, 1].     

Averaged Pointwise Bounds for Deformations of Schrödinger Eigenfunctions

Let (M,g) be an n-dimensional, compact Riemannian manifold and ${P_0(\hbar) = -\hbar{^2} \Delta_g + V(x)}$ be a semiclassical Schrödinger operator with ${\hbar \in (0,\hbar_0]}$ . Let ${E(\hbar) \in [E-o(1),E+o(1)]}$ and ${(\phi_{\hbar})_{\hbar \in (0,\hbar_0]}}$ be a family of L ²-normalized eigenfunctions of ${P_0(\hbar)}$ with ${P_0(\hbar) \phi_{\hbar} = E(\hbar) \phi_{\hbar}}$ . We consider magnetic deformations of ${P_0(\hbar)}$ of the form ${P_u(\hbar) = - \Delta_{\omega_u}(\hbar) + V(x)}$ , where ${\Delta_{\omega_u}(\hbar) = (\hbar d + i \omega_u(x))^*({\hbar}d + i \omega_u(x))}$ . Here, u is a k-dimensional parameter running over ${B^k(\epsilon)}$ (the ball of radius ${\epsilon}$ ), and the family of the magnetic potentials ${(w_u)_{u\in B^k(\epsilon)}}$ satisfies the admissibility condition given in Definition 1.1. This condition implies that k ≥ n and is generic under this assumption. Consider the corresponding family of deformations of ${(\phi_{\hbar})_{\hbar \in (0, \hbar_0]}}$ , given by ${(\phi^u_{\hbar})_{\hbar \in(0, \hbar_0]}}$ , where $$\phi_{\hbar}^{(u)}:= {\rm e}^{-it_0 P_u(\hbar)/\hbar}\phi_{\hbar}$$ for ${|t_0|\in (0,\epsilon)}$ ; the latter functions are themselves eigenfunctions of the ${\hbar}$ -elliptic operators ${Q_u(\hbar): ={\rm e}^{-it_0P_u(\hbar)/\hbar} P_0(\hbar) {\rm e}^{it_0 P_u(\hbar)/\hbar}}$ with eigenvalue ${E(\hbar)}$ and ${Q_0(\hbar) = P_{0}(\hbar)}$ . Our main result, Theorem1.2, states that for ${\epsilon >0 }$ small, there are constants ${C_j=C_j(M,V,\omega,\epsilon) > 0}$ with j = 1,2 such that $$C_{1}\leq \int\limits_{\mathcal{B}^k(\epsilon)} |\phi_{\hbar}^{(u)}(x)|^2 \, {\rm d}u \leq C_{2}$$ , uniformly for ${x \in M}$ and ${\hbar \in (0,h_0]}$ . We also give an application to eigenfunction restriction bounds in Theorem 1.3.     

Global injectivity conditions for planar maps

We prove that a planar $C^1$ -smooth map $f:D\longrightarrow \mathbb{R }^{2n}$ , where $D\subseteq \mathbb{R }^{2n}$ is a convex open set, is injective if $\mathbb{R }\cap \mathrm{Spec}(df)_z=\emptyset $ for all $z\in D$ . We continue by showing that the triangulability of the differentials $(df)_z$ , $z\in D$ , ensure the global injectivity as well.     

Properties of finite unrefinable chains of ring topologies

Let R(+, ·) be a nilpotent ring and $ \left( {\mathfrak{M}, < } \right) $ be the lattice of all ring topologies on R(+, ·) or the lattice of all such ring topologies on R(+, ·) in each of which the ring R possesses a basis of neighborhoods of zero consisting of subgroups. Let ?? and ??? be ring topologies from $ \mathfrak{M} $ such that $ \tau = {\tau_0}{ \prec_\mathfrak{M}}{\tau_1}{ \prec_\mathfrak{M}} \cdots { \prec_\mathfrak{M}}{\tau_n} = \tau ^{\prime} $ . Then k????n for every chain $ \tau = {\tau ^{\prime}_0} < {\tau ^{\prime}_1} < \cdots < {\tau ^{\prime}_k} = \tau ^{\prime} $ of topologies from $ \mathfrak{M} $ , and also n?=?k if and only if $ {\tau ^{\prime}_i}{ \prec_\mathfrak{M}}{\tau ^{\prime}_{i + 1}} $ for all 0????i?<?k.