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1

Resistance forms, quasisymmetric maps, and heat kernel estimates. American Mathematical Society, 2012.

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2

Weber, Andreas. Heat kernel estimates and L_1hnp spectral theory of locally symmetric spaces. Univ.-Verl. Karlsruhe, 2006.

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3

Volume doubling measures and heat kernel estimates on self-similar sets. American Mathematical Society, 2009.

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4

Barlow, M. T. Characterization of sub-Gaussian heat kernel estimates on strongly recurrent graphs. Kyōto Daigaku Sūri Kaiseki Kenkyūjo, 2004.

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5

Kumagai, Takashi. Heat kernel estimates and parabolic Harnack imequalities on graphs and resistance forms. Kyōto Daigaku Sūri Kaiseki Kenkyūjo, 2003.

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6

Fujii, Ichiro. Heat kernel estimates on the incipient infinite cluster for critical branching processes. Kyōto Daigaku Sūri Kaiseki Kenkyūjo, 2007.

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7

Chen, Zhen-Qing. Heat Kernel Estimates for Jump Processes of Mixed Types on Metric Measure Spaces. Research Institute for Mathematical Sciences, Kyoto University, 2006.

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8

Barlow, M. T. Parabolic harnack inequality and heat kernel estimates for random walks with long range fumps. Kyōto Daigaku Sūri Kaiseki Kenkyūjo, 2007.

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9

Jenkins, Stephen P. Did the middle class shrink during the 1980's?: UK evidence from kernel density estimates. ESRC Research Centre on Micro-social Change, 1995.

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10

Newey, Whitney K. Kernel estimation of partial means and a general variance estimator. Dept. of Economics, Massachusetts Institute of Technology, 1992.

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11

Jong, Robert M. de. Consistency of kernel estimators of heteroscedastic and autocorrelated covariance matrices. Cardiff Business School, 1996.

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12

Ferraty, Frédéric, and Philippe Vieu. Kernel Regression Estimation for Functional Data. Edited by Frédéric Ferraty and Yves Romain. Oxford University Press, 2018. http://dx.doi.org/10.1093/oxfordhb/9780199568444.013.4.

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This article provides an overview of recent nonparametric and semiparametric advances in kernel regression estimation for functional data. In particular, it considers the various statistical techniques based on kernel smoothing ideas that have recently been developed for functional regression estimation problems. The article first examines nonparametric functional regression modelling before discussing three popular functional regression estimates constructed by means of kernel ideas, namely: the Nadaraya-Watson convolution kernel estimate, the kNN functional estimate, and the local linear fun
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13

Epstein, Charles L., and Rafe Mazzeo. Holder Estimates for the 1-dimensional Model Problems. Princeton University Press, 2017. http://dx.doi.org/10.23943/princeton/9780691157122.003.0006.

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This chapter establishes Hölder space estimates for the 1-dimensional model problems. It gives a detailed treatment of the 1-dimensional case, in part because all of the higher dimensional estimates are reduced to estimates on heat kernels for the 1-dimensional model problems. It also presents the proof of parabolic Schauder estimates for the generalized Kimura diffusion operator using the explicit formula for the heat kernel, along with standard tools of analysis. Finally, it considers kernel estimates for degenerate model problems, explains how Hölder estimates are obtained for the 1-dimensi
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14

Epstein, Charles L., and Rafe Mazzeo. Holder Estimates for Euclidean Models. Princeton University Press, 2017. http://dx.doi.org/10.23943/princeton/9780691157122.003.0008.

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This chapter presents the Hölder space estimates for Euclidean model problems. It first considers the homogeneous Cauchy problem and the inhomogeneous problem before defining the resolvent operator as the Laplace transform of the heat kernel. It then describes the 1-dimensional kernel estimates that form essential components of the proofs of the Hölder estimates for the general model problems; these include basic kernel estimates, first derivative estimates, and second derivative estimates. The proofs of these estimates are elementary. The chapter concludes by proving estimates on the resolven
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15

Epstein, Charles L., and Rafe Mazzeo. Holder Estimates for Higher Dimensional Corner Models. Princeton University Press, 2017. http://dx.doi.org/10.23943/princeton/9780691157122.003.0007.

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This chapter establishes Hölder space estimates for higher dimensional corner model problems. It first explains the homogeneous Cauchy problem before estimating the solution of the inhomogeneous problem in a n-dimensional corner. It then reduces the proof of an estimate in higher dimensions to the estimation of a product of 1-dimensional integrals. Using the “1-variable-at-a-time” method, the chapter proves the higher dimensional estimates in several stages by considering the “pure corner” case where m = 0, and then turns to the Euclidean case, where n = 0. It also discusses the resolvent oper
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16

Epstein, Charles L., and Rafe Mazzeo. Holder Estimates for General Models. Princeton University Press, 2017. http://dx.doi.org/10.23943/princeton/9780691157122.003.0009.

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This chapter presents the Hölder estimates for general model problems. It first estimates solutions to heat equations for both the homogeneous Cauchy problem and the inhomogeneous problem, obtaining first and second derivative estimates in the latter case, before discussing a general result describing the off-diagonal and long-time behavior of the solution kernel for the general model. It also states a proposition summarizing the properties of the resolvent operator as an operator on the Hölder spaces. In contrast to the case of the heat equation, there is no need to assume that the data has c
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17

Epstein, Charles L., and Rafe Mazzeo. The Resolvent Operator. Princeton University Press, 2017. http://dx.doi.org/10.23943/princeton/9780691157122.003.0011.

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This chapter describes the construction of a resolvent operator using the Laplace transform of a parametrix for the heat kernel and a perturbative argument. In the equation (μ‎-L) R(μ‎) f = f, R(μ‎) is a right inverse for (μ‎-L). In Hölder spaces, these are the natural elliptic estimates for generalized Kimura diffusions. The chapter first constructs the resolvent kernel using an induction over the maximal codimension of bP, and proves various estimates on it, along with corresponding estimates for the solution operator for the homogeneous Cauchy problem. It then considers holomorphic semi-gro
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18

V, Burkhauser Richard, and ESRC Research Centre on Micro-social Change., eds. Where in the world is the middle class?: A cross-national comparison of the vanishing middle class using kernel density estimates. ESRC Research Centre on Micro-social Change, 1996.

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19

Epstein, Charles L., and Rafe Mazzeo. The Model Solution Operators. Princeton University Press, 2017. http://dx.doi.org/10.23943/princeton/9780691157122.003.0004.

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This chapter introduces the model problems and the solution operator for the associated heat equations. These operators give a good approximation for the behavior of the heat kernel in neighborhoods of different types of boundary points. The chapter states and proves the elementary features of these operators and shows that the model heat operators have an analytic continuation to the right half plane. It first considers the model problem in 1-dimension and in higher dimensions before discussing the solution to the homogeneous Cauchy problem. It then describes the first steps toward perturbati
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20

Delsol, Laurent. Nonparametric Methods for α-Mixing Functional Random Variables. Редактори Frédéric Ferraty та Yves Romain. Oxford University Press, 2018. http://dx.doi.org/10.1093/oxfordhb/9780199568444.013.5.

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This article considers how functional kernel methods can be used to study α-mixing datasets. It first provides an overview of how prediction problems involving dependent functional datasets may arise from the study of time series, focusing on the standard discretized model and modelization that takes into account the functional nature of the evolution of the quantity to be studied over time. It then considers strong mixing conditions, with emphasis on the notion of α-mixing coefficients and α-mixing variables introduced by Rosenblatt (1956). It also describes some conditions for a Markov chain
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21

Zhu, Yang, and Miroslav Krstic. Delay-Adaptive Linear Control. Princeton University Press, 2020. http://dx.doi.org/10.23943/princeton/9780691202549.001.0001.

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Actuator and sensor delays are among the most common dynamic phenomena in engineering practice, and when disregarded, they render controlled systems unstable. Over the past sixty years, predictor feedback has been a key tool for compensating such delays, but conventional predictor feedback algorithms assume that the delays and other parameters of a given system are known. When incorrect parameter values are used in the predictor, the resulting controller may be as destabilizing as without the delay compensation. This book develops adaptive predictor feedback algorithms equipped with online est
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