Compact Embedding Research Articles

Embeddings of the fractional Sobolev spaces on metric-measure spaces

We study embeddings of fractional Sobolev spaces defined on metric-measure spaces. Various results about continuous and compact embeddings are proven. Many theorems are illustrated by a number of examples.

Poincaré inequalities and compact embeddings from Sobolev type spaces into weighted Lq spaces on metric spaces

We study compactness and boundedness of embeddings from Sobolev type spaces on metric spaces into Lq spaces with respect to another measure. The considered Sobolev spaces can be of fractional order and some statements allow also nondoubling measures. Our results are formulated in a general form, using sequences of covering families and local Poincaré type inequalities. We show how to construct such suitable coverings and Poincaré inequalities. For locally doubling measures, we prove a self-improvement property for two-weighted Poincaré inequalities, which applies also to lower-dimensional measures.We simultaneously treat various Sobolev spaces, such as the Newtonian, fractional Hajłasz and Poincaré type spaces, for rather general measures and sets, including fractals and domains with fractal boundaries. By considering lower-dimensional measures on the boundaries of such domains, we obtain trace embeddings for the above spaces. In the case of Newtonian spaces we exactly characterize when embeddings into Lq spaces with respect to another measure are compact. Our tools are illustrated by concrete examples. For measures satisfying suitable dimension conditions, we recover several classical embedding theorems on domains and fractal sets in Rn.

On a new fractional Sobolev space with variable exponent on complete manifolds

We present the theory of a new fractional Sobolev space in complete manifolds with variable exponent. As a result, we investigate some of our new space’s qualitative properties, such as completeness, reflexivity, separability, and density. We also show that continuous and compact embedding results are valid. We apply the conclusions of this study to the variational analysis of a class of fractional p(z, cdot )-Laplacian problems involving potentials with vanishing behavior at infinity as an application.

Structural reflection, shrewd cardinals and the size of the continuum

Motivated by results of Bagaria, Magidor and Väänänen, we study characterizations of large cardinal properties through reflection principles for classes of structures. More specifically, we aim to characterize notions from the lower end of the large cardinal hierarchy through the principle [Formula: see text] introduced by Bagaria and Väänänen. Our results isolate a narrow interval in the large cardinal hierarchy that is bounded from below by total indescribability and from above by subtleness, and contains all large cardinals that can be characterized through the validity of the principle [Formula: see text] for all classes of structures defined by formulas in a fixed level of the Lévy hierarchy. Moreover, it turns out that no property that can be characterized through this principle can provably imply strong inaccessibility. The proofs of these results rely heavily on the notion of shrewd cardinals, introduced by Rathjen in a proof-theoretic context, and embedding characterizations of these cardinals that resembles Magidor’s classical characterization of supercompactness. In addition, we show that several important weak large cardinal properties, like weak inaccessibility, weak Mahloness or weak [Formula: see text]-indescribability, can be canonically characterized through localized versions of the principle [Formula: see text]. Finally, the techniques developed in the proofs of these characterizations also allow us to show that Hamkin’s weakly compact embedding property is equivalent to Lévy’s notion of weak [Formula: see text]-indescribability.

Nontrivial solutions for a class of Hamiltonian strongly degenerate elliptic system

In this paper, we study the existence of nontrivial to the following Hamiltonian strongly degenerate elliptic system in whole space where are nonnegative functions with and V might unbounded or vanish at infinity. Here is the strongly degenerate operator and the nonlinearity terms are continuous functions and satisfy some assumptions that will be specified later. Due to the unbounded or vanishing of potentials and degeneracy of the operator, some new compact embedding theorems are used in the proof. Our results extend and generalize some existing results.

Hilbert complexes with mixed boundary conditions part 1: de Rham complex

We show that the de Rham Hilbert complex with mixed boundary conditions on bounded strong Lipschitz domains is closed and compact. The crucial results are compact embeddings which follow by abstract arguments using functional analysis together with particular regular decompositions. Higher Sobolev order results are proved as well.

Compactness of nonlinear integral operators with discontinuous and with singular kernels

This paper establishes compactness of nonlinear integral operators in the space of continuous functions. One result deals with operators whose kernel can have jumps across a finite number of curves, which typically arise from the study of ordinary differential equations with boundary conditions of local or nonlocal type. Several other results deal with operators whose kernels have a singularity, which arise from the study of fractional differential equations. We motivate the study of these integral equations by discussing some initial value problems for fractional differential equations of Caputo and Riemann-Liouville type. We prove a compact embedding theorem for fractional integrals in order to give a new treatment for the singular kernel case.

Global well-posedness and critical norm concentration for inhomogeneous biharmonic NLS

We consider the inhomogeneous biharmonic nonlinear Schr dinger (IBNLS) equation in $${\mathbb {R}}^N$$ , $$\begin{aligned} i \partial _t u +\Delta ^2 u -|x|^{-b} |u|^{2\sigma }u = 0, \end{aligned}$$ where $$\sigma > 0$$ and $$b > 0$$ . We first study the local well-posedness in $${\dot{H}}^{s_c}\cap \dot{H}^2 $$ , for $$N\ge 5$$ and $$0<s_c<2$$ , where $$s_c=\frac{N}{2}-\frac{4-b}{2\sigma }$$ . Next, we established a Gagliardo-Nirenberg type inequality in order to obtain sufficient conditions for global existence of solutions in $$\dot{H}^{s_c}\cap \dot{H}^2$$ with $$0\le s_c<2$$ . Finally, we study the phenomenon of $$L^{\sigma _c}$$ -norm concentration for finite time blow up solutions with bounded $$\dot{H}^{s_c}$$ -norm, where $$\sigma _c=\frac{2N\sigma }{4-b}$$ . Our main tool is the compact embedding of $$\dot{L}^p\cap \dot{H}^2$$ into a weighted $$L^{2\sigma +2}$$ space, which may be seen of independent interest.

RoBERTa-LSTM: A Hybrid Model for Sentiment Analysis With Transformer and Recurrent Neural Network

Due to the rapid development of technology, social media has become more and more common in human daily life. Social media is a platform for people to express their feelings, feedback, and opinions. To understand the sentiment context of the text, sentiment analysis plays the role to determine whether the sentiment of the text is positive, negative, neutral or any other personal feeling. Sentiment analysis is prominent from the perspective of business or politics where it highly impacts the strategic decision making. The challenges of sentiment analysis are attributable to the lexical diversity, imbalanced dataset and long-distance dependencies of the texts. In view of this, a data augmentation technique with GloVe word embedding is leveraged to synthesize more lexically diverse samples by similar word vector replacements. The data augmentation also focuses on the oversampling of the minority classes to mitigate the imbalanced dataset problems. Apart from that, the existing sentiment analysis mostly leverages sequence models to encode the long-distance dependencies. Nevertheless, the sequence models require a longer execution time as the processing is done sequentially. On the other hand, the Transformer models require less computation time with parallelized processing. To that end, this paper proposes a hybrid deep learning method that combines the strengths of sequence model and Transformer model while suppressing the limitations of sequence model. Specifically, the proposed model integrates Robustly optimized BERT approach and Long Short-Term Memory for sentiment analysis. The Robustly optimized BERT approach maps the words into a compact meaningful word embedding space while the Long Short-Term Memory model captures the long-distance contextual semantics effectively. The experimental results demonstrate that the proposed hybrid model outshines the state-of-the-art methods by achieving F1-scores of 93&#x0025;, 91&#x0025;, and 90&#x0025; on IMDb dataset, Twitter US Airline Sentiment dataset, and Sentiment140 dataset, respectively.

Dynamics of stochastic retarded Benjamin-Bona-Mahony equations on unbounded channels

&lt;p style='text-indent:20px;'&gt;This article is devoted to the asymptotic behaviour of solutions for stochastic Benjamin-Bona-Mahony (BBM) equations with distributed delay defined on unbounded channels. We first prove the existence, uniqueness and forward compactness of pullback random attractors (PRAs). We then establish the forward asymptotic autonomy of this PRA. Finally, we study the non-delay stability of this PRA. Due to the loss of usual compact Sobolev embeddings on unbounded domains, the forward uniform tail-estimates and forward flattening of solutions are used to prove the forward asymptotic compactness of solutions.&lt;/p&gt;

The Long-Term Dynamic Behavior of Solutions to a Class of Generalized Higher-Order Kirchhoff-Type Coupled Wave Equations

In this paper, we study the long-term dynamic behavior of a class of generalized high-order Kirchhoff-type coupled wave equations. Firstly, the existence of uniqueness global solution of this kind of equations in Ek space is proved by prior estimation and Galerkin method; Then, through using Rellich-Kondrachov compact embedding theorem, it is proved that the solution semigroup S(t) has the family of the global attractors Ak in space Ek; Finally, through linearization method, proves that the operator semigroup S(t) Frechet differentiable and the attenuation of linearization problem volume element. Furthermore, we can obtain the finite Hausdorff dimension and Fractal dimension of the family of the global attractors Ak.

A uniform bound on costs of controlling semilinear heat equations on a sequence of increasing domains and its application

In this paper, we first prove a uniform upper bound on costs of null controls for semilinear heat equations with globally Lipschitz nonlinearity on a sequence of increasing domains, where the controls are acted on an equidistributed set that spreads out in the whole Euclidean space ℝN. As an application, we then show the exact null-controllability for this semilinear heat equation in ℝN. The main novelty here is that the upper bound on costs of null controls for such kind of equations in large but bounded domains can be made uniformly with respect to the sizes of domains under consideration. The latter is crucial when one uses a suitable approximation argument to derive the global null-controllability for the semilinear heat equation in ℝN. This allows us to overcome the well-known problem of the lack of compactness embedding arising in the study of null-controllability for nonlinear PDEs in generally unbounded domains.

Some compact and non-compact embedding theorems for the function spaces defined by fractional Fourier transform

The fractional Fourier transform is a generalization of the classical Fourier transform through an angular parameter $\alpha $. This transform uses in quantum optics and quantum wave field reconstruction, also its application provides solving some differrential equations which arise in quantum mechanics. The aim of this work is to discuss compact and non-compact embeddings between the spaces $A_{\alpha ,p}^{w,\omega }\left(\mathbb{R}^{d}\right) $ which are the set of functions in ${L_{w}^{1}\left(\mathbb{R}^{d}\right) }$ whose fractional Fourier transform are in ${L_{\omega}^{p}\left(\mathbb{R}^{d}\right) }$. Moreover, some relevant counterexamples are indicated.

View-Invariant, Occlusion-Robust Probabilistic Embedding for Human Pose

Recognition of human poses and actions is crucial for autonomous systems to interact smoothly with people. However, cameras generally capture human poses in 2D as images and videos, which can have significant appearance variations across viewpoints that make the recognition tasks challenging. To address this, we explore recognizing similarity in 3D human body poses from 2D information, which has not been well-studied in existing works. Here, we propose an approach to learning a compact view-invariant embedding space from 2D body joint keypoints, without explicitly predicting 3D poses. Input ambiguities of 2D poses from projection and occlusion are difficult to represent through a deterministic mapping, and therefore we adopt a probabilistic formulation for our embedding space. Experimental results show that our embedding model achieves higher accuracy when retrieving similar poses across different camera views, in comparison with 3D pose estimation models. We also show that by training a simple temporal embedding model, we achieve superior performance on pose sequence retrieval and largely reduce the embedding dimension from stacking frame-based embeddings for efficient large-scale retrieval. Furthermore, in order to enable our embeddings to work with partially visible input, we further investigate different keypoint occlusion augmentation strategies during training. We demonstrate that these occlusion augmentations significantly improve retrieval performance on partial 2D input poses. Results on action recognition and video alignment demonstrate that using our embeddings without any additional training achieves competitive performance relative to other models specifically trained for each task.

On inclusion properties of discrete Morrey spaces

Abstract We discuss a necessary condition for inclusion relations of weak type discrete Morrey spaces which can be seen as an extension of the results in [H. Gunawan, E. Kikianty and C. Schwanke, Discrete Morrey spaces and their inclusion properties, Math. Nachr. 291 2018, 8–9, 1283–1296] and [D. D. Haroske and L. Skrzypczak, Morrey sequence spaces: Pitt’s theorem and compact embeddings, Constr. Approx. 51 2020, 3, 505–535]. We also prove a proper inclusion from weak type discrete Morrey spaces into discrete Morrey spaces. In addition, we give a necessary condition for this inclusion. Some connections between the inclusion properties of discrete Morrey spaces and those of Morrey spaces are also discussed.

Long-time dynamics of fractional nonclassical diffusion equations with nonlinear colored noise and delay on unbounded domains

This paper deals with the long-time dynamics for a class of highly nonlinear fractional nonclassical diffusion equations with nonlinear colored noise and time delay defined on the whole space Rn. The existence and uniqueness of tempered pullback random attractors of the equations are established in C([−ρ,0],Hα(Rn)) (ρ>0 and α∈(0,1)) for polynomial growth drift and diffusion terms as well as Lipschitz time-delay terms. In a special case where the nonlinear diffusion term depends only on the space variable, the approximation of those random attractors is also investigated in C([−ρ,0],Hα(Rn)) when the correlation time of the colored approaches zero. The pullback asymptotical compactness of the solutions in C([−ρ,0],Hα(Rn)) is proved by virtue of the arguments of Arzela-Ascoli theorem, spectral decomposition as well as uniform tail-estimates developed by Wang (1999) [59] in order to surmount several difficulties caused by the lack of compact Sobolev embeddings on unbounded domains as well as the weakly dissipative distinguishing structures of the equations.

Compact Embedding Theorems for The Space of Functions with Wavelet Transform in Amalgam Spaces

Bu çalışma dalgacık dönüşümü kullanarak 〖 A〗_s (W)_(ω,ϑ)^(p,q,r) (R) uzayını tanımlamak ve ayrıca bu uzayda kapsama, kompakt gömülme teoremlerini incelemek için motive edilmiştir.

Deep embeddings and logistic regression for rapid active learning in histopathological images

Background and ObjectiveRecognizing different tissue components is one of the most fundamental and essential works in digital pathology. Current methods are often based on convolutional neural networks (CNNs), which need numerous annotated samples for training. Creating large-scale histopathological datasets is labor-intensive, where interactive data annotation is a potential solution. MethodsWe propose DELR (Deep Embedding-based Logistic Regression) to enable rapid model training and inference for histopathological image analysis. DELR utilizes a pretrained CNN to encode images as compact embeddings with low computational cost. The embeddings are then used to train a Logistic Regression model efficiently. We implemented DELR in an active learning framework, and validated it on three histopathological problems (binary, 4-category, and 8-category classification challenge for lung, breast, and colorectal cancer, respectively). We also investigated the influence of active learning strategy and type of the encoder. ResultsOn all the three datasets, DELR can achieve an area under curve (AUC) metric higher than 0.95 with only 100 image patches per class. Although its AUC is slightly lower than a fine-tuned CNN counterpart, DELR can be 536, 316, and 1481 times faster after pre-encoding. Moreover, DELR is proved to be compatible with a variety of active learning strategies and encoders. ConclusionsDELR can achieve comparable accuracy to CNN with rapid running speed. These advantages make it a potential solution for real-time interactive data annotation.

Nuclear embeddings in general vector-valued sequence spaces with an application to Sobolev embeddings of function spaces on quasi-bounded domains

We study nuclear embeddings for spaces of Besov and Triebel-Lizorkin type defined on quasi-bounded domains Ω⊂Rd. The counterpart for such function spaces defined on bounded domains has been considered for a long time and the complete answer was obtained only recently. Compact embeddings for function spaces defined on quasi-bounded domains have already been studied in detail, also concerning their entropy and s-numbers. We now prove the first and complete nuclearity result in this context. The concept of nuclearity has already been introduced by Grothendieck in 1955. Our second main contribution is the generalisation of the famous Tong result (1969) which characterises nuclear diagonal operators acting between sequence spaces of ℓr type, 1≤r≤∞. We can now extend this to the setting of general vector-valued sequence spaces of type ℓq(βjℓpMj) with 1≤p,q≤∞, Mj∈N0 and a weight sequence with βj>0. In particular, we prove a criterion for the embedding idβ:ℓq1(βjℓp1Mj)↪ℓq2(ℓp2Mj) to be nuclear.

Compact Sobolev-Slobodeckij embeddings and positive solutions to fractional Laplacian equations

AbstractIn this work, we study the existence of a positive solution to an elliptic equation involving the fractional Laplacian (−Δ)sin ℝn, forn≥ 2, such as(0.1)(−Δ)su+E(x)u+V(x)uq−1=K(x)f(u)+u2s⋆−1.$$(-\Delta)^{s} u+E(x) u+V(x) u^{q-1}=K(x) f(u)+u^{2_{s}^{\star}-1}.$$Here,s ∈(0, 1),q∈2,2s⋆$q \in\left[2,2_{s}^{\star}\right)$with2s⋆:=2nn−2s$2_{s}^{\star}:=\frac{2 n}{n-2 s}$being the fractional critical Sobolev exponent,E(x),K(x),V(x) &gt; 0 : ℝn→ ℝ are measurable functions which satisfy joint “vanishing at infinity” conditions in a measure-theoretic sense, andf(u) is a continuous function on ℝ of quasi-critical, super-q-linear growth withf(u) ≥ 0 ifu≥ 0. Besides, we study the existence of multiple positive solutions to an elliptic equation in ℝnsuch as(0.2)(−Δ)su+E(x)u+V(x)uq−1=λK(x)ur−1,$$(-\Delta)^{s} u+E(x) u+V(x) u^{q-1}=\lambda K(x) u^{r-1},$$where 2 &lt;r&lt;q&lt; ∞(both possibly (super-)critical),E(x),K(x),V(x) &gt; 0 : ℝn→ ℝ are measurable functions satisfying joint integrability conditions, andλ&gt; 0 is a parameter. To study (0.1)-(0.2), we first describe a family of general fractional Sobolev-Slobodeckij spacesMs;q,p(ℝn) as well as their associated compact embedding results.