Lipschitz continuity, global smooth approximations and extension theorems for Sobolev functions in Carnot-Carathéodory spaces

Garofalo, Nicola; Nhieu, Duy-Minh

doi:10.1007/bf02819446

Cited by 133 publications

(132 citation statements)

References 23 publications

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“…For every i sufficiently big U cc,φ i (x, r/2) ⊂ ω. Moreover, by [28,Theorem 2.7], ∀x, y ∈ U cc,φ i (x, r/2) Viceversa, by definition of differential it follows that ∇ φ i φ i L ∞ (U cc,φ i (x,r/2))) ≤ Lip cc (φ i , U cc,φ i (x, r/2)) ∀i ∈ N.…”

Section: Applicationmentioning

confidence: 99%

Smooth approximation for intrinsic Lipschitz functions in the Heisenberg group

et al. 2013

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Section: Applicationmentioning

confidence: 99%

Smooth approximation for intrinsic Lipschitz functions in the Heisenberg group

et al. 2013

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“…, X m ) on M, see [41], [3] and [23]. Following [3] and [24] our first standing hypothesis is that (1.2) the inclusion i : (R n , | · |) → (R n , d) is continuous.…”

Section: Introduction and Statement Of Main Resultsmentioning

confidence: 99%

“…1 For a detailed study on the validity of "H = W " in general metric measure spaces, and in particular on the relation between the definitions used in this paper and the more commonly used definition based on the closure of the class of smooth functions with compact support see [22], [31], [24], [20], [21] and [43].…”

Section: Introduction and Statement Of Main Resultsmentioning

confidence: 99%

Harnack estimates for degenerate parabolic equations modeled on the subelliptic p -Laplacian

Avelin

Capogna

Citti

et al. 2014

Advances in Mathematics

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Abstract. We establish a Harnack inequality for a class of quasi-linear PDE modeled on the prototypewhere p ≥ 2, X = (X 1 , . . . , Xm) is a system of Lipschitz vector fields defined on a smooth manifold M endowed with a Borel measure µ, and X * i denotes the adjoint of X i with respect to µ. Our estimates are derived assuming that (i) the control distance d generated by X induces the same topology on M; (ii) a doubling condition for the µ-measure of d−metric balls and (iii) the validity of a Poincaré inequality involving X and µ. Our results extend the recent work in [15], [35], to a more general setting including the model cases of (1) metrics generated by Hörmander vector fields and Lebesgue measure; (2) Riemannian manifolds with non-negative Ricci curvature and Riemannian volume forms; and (3) metrics generated by non-smooth Baouendi-Grushin type vector fields and Lebesgue measure. In all cases the Harnack inequality continues to hold when the Lebesgue measure is substituted by any smooth volume form or by measures with densities corresponding to Muckenhoupt type weights.2000 Mathematics Subject Classification.

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“…The derivative function is obviously continuous and differentiable, so it is locally Lipschitz continuous [20]. According to this,…”

Section: Optimization Algorithmmentioning

confidence: 99%

Predict Quit Rate of Group from User Behavioral and Social Information

Tu¹,

Cui²

2015

IJKE

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Abstract-There is a common intuition that the user behavioral pattern and social information of a group may influence its attraction to users. In this paper, we employ user behavioral and social information to predict the user quit rate of social groups and validate the link between social behavioral pattern and group quit rate on a large scale real datasetTencent QQ groups. We routinely model this task as a regression problem, and generate 97 features from user behavioral and social information. Then we use an improved Scalable Orthogonal Regression (iSOR) method to predict the quit rate of QQ group. Our study shows that the quit rate of a group can be predicted with high accuracy, furthermore, the iSOR method selected several import features from the total 97 social behavioral features.Index Terms-Quit rate, social group, social information, user behavior. I. INTRODUCTIONUnderstanding the social and user behavioral information of groups has been an attractive topic in social network analysis. Most of the existing works have focused on the social community detection (e.g.[1]- [3]), group profiling (e.g.[4], [5]) and group stability analysis (e.g. [6]-[10]) using social behavioral information. But little attention is paid to the link between social behavioral pattern and user quit rate of a group, which has great significance to both social science and the industry. For social science, it helps to understand the human collective behaviors. For industry, the social network operators can serve the customers more intelligently. In this paper, we extract 97 features for quit rate prediction and we explore the relation between social behavioral information and quit rate of a group.There exist various forms of groups in many online social network services [11]. In Tencent QQ groups, for example, user can find groups of her interest by searching keywords or group ID. She can join a group when permitted by the group manager or just quit a group she joined without any restriction. All these groups are created and maintained by QQ users. Users in a group can send messages that all group members can see. The continuous records of the user behavioral and social information in a group provide us the opportunity to study the relation between social behavioral information and quit rate of a group.In this paper, we extract lots of features from social behavioral information and predict the quit rate using an improved linear regression model, which could select significant orthogonal features automatically [12]. Our experiment result shows that the quit rate of a group can be predicted precisely using social behavioral pattern without knowing the conversation content, which validates the link between social behavioral information and group attraction. We also show that the behavioral pattern of group conversation is important to attract users staying in a group.The main contributions of this paper are: 1) Extract useful social features and user behavioral features for quit rate prediction. We totally extract 97 features from use...

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Lipschitz continuity, global smooth approximations and extension theorems for Sobolev functions in Carnot-Carathéodory spaces

Cited by 133 publications

References 23 publications

Smooth approximation for intrinsic Lipschitz functions in the Heisenberg group

Smooth approximation for intrinsic Lipschitz functions in the Heisenberg group

Harnack estimates for degenerate parabolic equations modeled on the subelliptic p -Laplacian

Predict Quit Rate of Group from User Behavioral and Social Information

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