Two weight inequality for vector-valued positive dyadic operators by parallel stopping cubes

We study the vector-valued positive dyadic operator $$T_\lambda(f\sigma):=\sum_{Q\in\mathcal{D}} \lambda_Q \int_Q f \mathrm{d}\sigma 1_Q,$$ where the coefficients $\{\lambda_Q:C\to D\}_{Q\in\mathcal{D}}$ are positive operators from a Banach lattice $C$ to a Banach lattice $D$. We assume that the Banach lattices $C$ and $D^*$ each have the Hardy--Littlewood property. An example of a Banach lattice with the Hardy--Littlewood property is a Lebesgue space. In the two-weight case, we prove that the $L^p_C(\sigma)\to L^q_D(\omega)$ boundedness of the operator $T_\lambda( \cdot \sigma)$ is characterized by the direct and the dual $L^\infty$ testing conditions: $$\lVert 1_Q T_\lambda(1_Q f \sigma)\rVert_{L^q_D(\omega)}\lesssim \lVert f\rVert_{L^\infty_C(Q,\sigma)} \sigma(Q)^{1/p},$$ $$\lVert1_Q T^*_{\lambda}(1_Q g \omega)\rVert_{L^{p'}_{C^*}(\sigma)}\lesssim \lVert g\rVert_{L^\infty_{D^*}(Q,\omega)} \omega(Q)^{1/q'}.$$ Here $L^p_C(\sigma)$ and $L^q_D(\omega)$ denote the Lebesgue--Bochner spaces associated with exponents $1<p\leq q<\infty$, and locally finite Borel measures $\sigma$ and $\omega$. In the unweighted case, we show that the $L^p_C(\mu)\to L^p_D(\mu)$ boundedness of the operator $T_\lambda( \cdot \mu)$ is equivalent to the endpoint direct $L^\infty$ testing condition: $$\lVert1_Q T_\lambda(1_Q f \mu)\rVert_{L^1_D(\mu)}\lesssim \lVert f\rVert_{L^\infty_C(Q,\mu)} \mu(Q).$$ This condition is manifestly independent of the exponent $p$. By specializing this to particular cases, we recover some earlier results in a unified way.Comment: 32 pages. The main changes are: a) Banach lattice-valued functions are considered. It is assumed that the Banach lattices have the Hardy--Littlewood property. b) The unweighted norm inequality is characterized by an endpoint testing condition and some corollaries of this characterization are stated. c) Some questions about the borderline of the vector-valued testing conditions are pose

Hierarchies from D-brane instantons in globally defined Calabi-Yau Orientifolds

We construct the first globally consistent semi-realistic Type I string vacua on an elliptically fibered manifold where the zero modes of the Euclidean D1-instanton sector allow for the generation of non-perturbative Majorana masses of an intermediate scale. In another class of global models, a D1-brane instanton can generate a Polonyi-type superpotential breaking supersymmetry at an exponentially suppressed scale.Comment: 4 pages, 4 tables, uses revtex; v2: Discussion of instanton curves improved, typos fixed, references added; v3: version published in PR

Influence of topological excitations on Shapiro steps and microwave dynamical conductance in bilayer exciton condensates

The quantum Hall state at total filling factor $\nu_T=1$ in bilayer systems realizes an exciton condensate and exhibits a zero-bias tunneling anomaly, similar to the Josephson effect in the presence of fluctuations. In contrast to conventional Josephson junctions, no Fraunhofer diffraction pattern has been observed, due to disorder induced topological defects, so-called merons. We consider interlayer tunneling in the presence of microwave radiation, and find Shapiro steps in the tunneling current-voltage characteristic despite the presence of merons. Moreover, the Josephson oscillations can also be observed as resonant features in the microwave dynamical conductance

Deep convolutional neural networks for estimating porous material parameters with ultrasound tomography

We study the feasibility of data based machine learning applied to ultrasound tomography to estimate water-saturated porous material parameters. In this work, the data to train the neural networks is simulated by solving wave propagation in coupled poroviscoelastic-viscoelastic-acoustic media. As the forward model, we consider a high-order discontinuous Galerkin method while deep convolutional neural networks are used to solve the parameter estimation problem. In the numerical experiment, we estimate the material porosity and tortuosity while the remaining parameters which are of less interest are successfully marginalized in the neural networks-based inversion. Computational examples confirms the feasibility and accuracy of this approach

From Stalemate to Deadlock : Clement’s Letter to Theodore in Recent Scholarship

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