Bounds on eigenvalues of perturbed Lamé operators with complex potentials

<abstract><p>Several recent papers have focused their attention in proving the correct analogue to the Lieb-Thirring inequalities for non self-adjoint operators and in finding bounds on the distribution of their eigenvalues in the complex plane. This paper provides some improvement in the state of the art in this topic. Precisely, we address the question of finding quantitative bounds on the discrete spectrum of the perturbed Lamé operator of elasticity $ -\Delta^\ast + V $ in terms of $ L^p $-norms of the potential. Original results within the self-adjoint framework are provided too.</p></abstract>

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Introduction: Research on the Self-Concept — the State of the Art in Eastern and Western Europe

Recent Research in Psychology - The Self-Concept ◽

10.1007/978-3-642-84143-9_1 ◽

1990 ◽

pp. 1-7 ◽

Cited By ~ 1

Author(s):

Louis Oppenheimer

Keyword(s):

Western Europe ◽

State Of The Art ◽

The State ◽

The Self ◽

Self Concept ◽

Eastern And Western Europe

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Collaborative Knowledge-Enhanced Recommendation with Self-Supervisions

Mathematics ◽

10.3390/math9172129 ◽

2021 ◽

Vol 9 (17) ◽

pp. 2129

Author(s):

Zhiqiang Pan ◽

Honghui Chen

Keyword(s):

Neural Network ◽

State Of The Art ◽

User Preferences ◽

The State ◽

The Self ◽

Knowledge Graph ◽

Model Optimization ◽

Interaction Graph ◽

Benchmark Datasets ◽

Collaborative Knowledge

Knowledge-enhanced recommendation (KER) aims to integrate the knowledge graph (KG) into collaborative filtering (CF) for alleviating the sparsity and cold start problems. The state-of-the-art graph neural network (GNN)–based methods mainly focus on exploiting the connectivity between entities in the knowledge graph, while neglecting the interaction relation between items reflected in the user-item interactions. Moreover, the widely adopted BPR loss for model optimization fails to provide sufficient supervisions for learning discriminative representation of users and items. To address these issues, we propose the collaborative knowledge-enhanced recommendation (CKER) method. Specifically, CKER proposes a collaborative graph convolution network (CGCN) to learn the user and item representations from the connection between items in the constructed interaction graph and the connectivity between entities in the knowledge graph. Moreover, we introduce the self-supervised learning to maximize the mutual information between the interaction- and knowledge-aware user preferences by deriving additional supervision signals. We conduct comprehensive experiments on two benchmark datasets, namely Amazon-Book and Last-FM, and the experimental results show that CKER can outperform the state-of-the-art baselines in terms of recall and NDCG on knowledge-enhanced recommendation.

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Practical Picture Processing

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100051700 ◽

1974 ◽

Vol 32 ◽

pp. 338-339

Author(s):

T. A. Welton

Keyword(s):

Radiation Damage ◽

Coherence Length ◽

Spatial Information ◽

State Of The Art ◽

Coherent Radiation ◽

The State ◽

Energy Spread ◽

Electron Micrograph ◽

Picture Processing ◽

Molecular Skeleton

Various authors have emphasized the spatial information resident in an electron micrograph taken with adequately coherent radiation. In view of the completion of at least one such instrument, this opportunity is taken to summarize the state of the art of processing such micrographs. We use the usual symbols for the aberration coefficients, and supplement these with £ and 6 for the transverse coherence length and the fractional energy spread respectively. He also assume a weak, biologically interesting sample, with principal interest lying in the molecular skeleton remaining after obvious hydrogen loss and other radiation damage has occurred.

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