scholarly journals Ancient Caloric Functions on Graphs With Unbounded Laplacians

2020 ◽  
Author(s):  
Bobo Hua
Keyword(s):  
Discrete Time ◽  
Continuous Time ◽  
Harmonic Functions ◽  
Polynomial Growth ◽  
Volume Growth ◽  
Heat Equations ◽  
Intrinsic Metric ◽  
Ancient Solutions

Abstract We study ancient solutions of polynomial growth to both continuous-time and discrete-time heat equations on graphs with unbounded Laplacians. We extend Colding and Minicozzi’s theorem [12] on manifolds and the result [22] on graphs with normalized Laplacians to the setting of graphs with unbounded Laplacians: for a graph admitting an intrinsic metric, which has polynomial volume growth, the dimension of the space of ancient solutions of polynomial growth is bounded by the dimension of harmonic functions with the same growth up to some factor.

scholarly journals Dimensional Bounds for Ancient Caloric Functions on Graphs

2019 ◽  
Author(s):  
Bobo Hua
Keyword(s):  
Harmonic Functions ◽  
Polynomial Growth ◽  
Volume Growth ◽  
Heat Equations ◽  
Ancient Solutions

Abstract We study ancient solutions of polynomial growth to heat equations on graphs and extend Colding and Minicozzi’s theorem [9] on manifolds to graphs: for a graph of polynomial volume growth, the dimension of the space of ancient solutions of polynomial growth is bounded by the product of the growth degree and the dimension of harmonic functions with the same growth.

scholarly journals Polynomial growth harmonic functions on groups of polynomial volume growth

2015 ◽  
Vol 280 (1-2) ◽  
pp. 551-567 ◽  
Author(s):  
Bobo Hua ◽  
Jürgen Jost
Keyword(s):  
Harmonic Functions ◽  
Polynomial Growth ◽  
Volume Growth

scholarly journals On the asymptotic behavior of the dimension of spaces of harmonic functions with polynomial growth

2020 ◽  
Vol 2020 (762) ◽  
pp. 281-306 ◽  
Author(s):  
Xian-Tao Huang
Keyword(s):  
Asymptotic Behavior ◽  
Harmonic Functions ◽  
Tangent Cone ◽  
Polynomial Growth ◽  
Volume Growth ◽  
Volume Ratio ◽  
Growth Order ◽  
Maximal Volume ◽  
Nonnegative Ricci Curvature ◽  
Asymptotic Volume

AbstractSuppose {(M^{n},g)} is a Riemannian manifold with nonnegative Ricci curvature, and let {h_{d}(M)} be the dimension of the space of harmonic functions with polynomial growth of growth order at most d. Colding and Minicozzi proved that {h_{d}(M)} is finite. Later on, there are many researches which give better estimates of {h_{d}(M)}. In this paper, we study the behavior of {h_{d}(M)} when d is large. More precisely, suppose {(M^{n},g)} has maximal volume growth and has a unique tangent cone at infinity. Then when d is sufficiently large, we obtain some estimates of {h_{d}(M)} in terms of the growth order d, the dimension n and the asymptotic volume ratio {\alpha=\lim_{R\rightarrow\infty}\frac{\mathrm{Vol}(B_{p}(R))}{R^{n}}}. When {\alpha=\omega_{n}}, i.e., {(M^{n},g)} is isometric to the Euclidean space, the asymptotic behavior obtained in this paper recovers a well-known asymptotic property of {h_{d}(\mathbb{R}^{n})}.

scholarly journals An almost rigidity theorem and its applications to noncompact RCD(0,N) spaces with linear volume growth

2018 ◽  
Vol 22 (04) ◽  
pp. 1850076 ◽  
Author(s):  
Xian-Tao Huang
Keyword(s):  
Distance Function ◽  
Metric Space ◽  
Level Sets ◽  
Harmonic Functions ◽  
Polynomial Growth ◽  
Volume Growth ◽  
Rigidity Theorem ◽  
Existence Problem ◽  
Maximal Volume ◽  
Almost Rigidity

The main results of this paper consist of two parts. First, we obtain an almost rigidity theorem which roughly says that on an [Formula: see text] space, when a domain between two level sets of a distance function has almost maximal volume compared to that of a cylinder, then this portion is close to a cylinder as a metric space. Second, we apply this almost rigidity theorem to study noncompact [Formula: see text] spaces with linear volume growth. More precisely, we obtain the sublinear growth of diameter of geodesic spheres, and study the non-existence problem of nonconstant harmonic functions with polynomial growth on such [Formula: see text] spaces.

Design of Programmable Wideband Low Pass Filter with Continuous-Time/Discrete-Time Hybrid Architecture

2017 ◽  
Vol E100.C (10) ◽  
pp. 858-865 ◽  
Author(s):  
Yohei MORISHITA ◽  
Koichi MIZUNO ◽  
Junji SATO ◽  
Koji TAKINAMI ◽  
Kazuaki TAKAHASHI
Keyword(s):  
Discrete Time ◽  
Continuous Time ◽  
Hybrid Architecture ◽  
Pass Filter ◽  
Low Pass Filter ◽  
Low Pass

scholarly journals Time to Intervene: A Continuous-Time Approach to Network Analysis and Centrality

Psychometrika ◽  
2021 ◽  
Author(s):  
Oisín Ryan ◽  
Ellen L. Hamaker
Keyword(s):  
Network Analysis ◽  
Discrete Time ◽  
Continuous Time ◽  
Centrality Measures ◽  
Time Interval ◽  
Direct Effects ◽  
Network Approach ◽  
Var Models ◽  
Auto Regressive ◽  
Conceptual Shift

AbstractNetwork analysis of ESM data has become popular in clinical psychology. In this approach, discrete-time (DT) vector auto-regressive (VAR) models define the network structure with centrality measures used to identify intervention targets. However, VAR models suffer from time-interval dependency. Continuous-time (CT) models have been suggested as an alternative but require a conceptual shift, implying that DT-VAR parameters reflect total rather than direct effects. In this paper, we propose and illustrate a CT network approach using CT-VAR models. We define a new network representation and develop centrality measures which inform intervention targeting. This methodology is illustrated with an ESM dataset.

On quasi-stationary distributions in absorbing continuous-time finite Markov chains

1967 ◽  
Vol 4 (1) ◽  
pp. 192-196 ◽  
Author(s):  
J. N. Darroch ◽  
E. Seneta
Keyword(s):  
Markov Chains ◽  
Discrete Time ◽  
Continuous Time ◽  
Present Note ◽  
Time Parameter ◽  
Stationary Distributions ◽  
Finite State ◽  
Absorbing Markov Chains ◽  
Finite Markov Chains ◽  
Finite State Space

In a recent paper, the authors have discussed the concept of quasi-stationary distributions for absorbing Markov chains having a finite state space, with the further restriction of discrete time. The purpose of the present note is to summarize the analogous results when the time parameter is continuous.

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