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# <title type="html">ArXiv Query: search_query=cat:math.CA AND ti:diffuse&id_list=&start=0&max_results=10</title>
# <id>http://arxiv.org/api/UbovMDQrOosYxZQzH/N9Sd9jTMY</id>
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# <entry>
# <id>http://arxiv.org/abs/1810.03952v2</id>
# <updated>2020-07-25T01:09:06Z</updated>
# <published>2018-10-09T13:19:21Z</published>
# <title>Fractional Diffusion Maps</title>
# <summary> In this paper, we extend the diffusion maps algorithm on a family of heat
# kernels that are either local (having exponential decay) or nonlocal (having
# polynomial decay), arising in various applications. For example, these kernels
# have been used as a regularizer in various supervised learning tasks for
# denoising images. Importantly, these heat kernels give rise to operators that
# include (but are not restricted to) the generators of the classical Laplacian
# associated to Brownian processes as well as the fractional Laplacian associated
# with $\beta$-stable L\'evy processes. For local kernels, while the method is a
# version of the diffusion maps algorithm, we show that the applications with
# non-Gaussian local heat kernels approximate temporally rescaled
# Laplace-Beltrami operators. For the non-local heat kernels, we modify the
# diffusion maps algorithm to estimate fractional Laplacian operators. Here, the
# graph distance is used to approximate the geodesic distance with appropriate
# error bounds. While this approximation becomes numerically expensive as the
# number of data points increases, it produces an accurate operator estimation
# that is robust to the choice of the kernel bandwidth parameter value. In
# contrast, the local kernels are numerically more efficient but more sensitive
# to the choice of kernel bandwidth parameter value. In an application to
# estimate non-smooth regression functions, we find that using the nonlocal
# kernel as a regularizer produces a more robust and accurate estimate than using
# local kernels. For manifolds with boundary, we find that the proposed
# fractional diffusion maps framework implemented with non-local kernels
# approximates the regional fractional Laplacian.
# </summary>
# <author>
# <name>Harbir Antil</name>
# </author>
# <author>
# <name>Tyrus Berry</name>
# </author>
# <author>
# <name>John Harlim</name>
# </author>
# <link href="http://arxiv.org/abs/1810.03952v2" rel="alternate" type="text/html"/>
# <link title="pdf" href="http://arxiv.org/pdf/1810.03952v2" rel="related" type="application/pdf"/>
# <arxiv:primary_category xmlns:arxiv="http://arxiv.org/schemas/atom" term="math.CA" scheme="http://arxiv.org/schemas/atom"/>
# <category term="math.CA" scheme="http://arxiv.org/schemas/atom"/>
# </entry>
# <entry>
# <id>http://arxiv.org/abs/1509.07707v1</id>
# <updated>2015-09-25T13:16:13Z</updated>
# <published>2015-09-25T13:16:13Z</published>
# <title>Iterated Diffusion Maps for Feature Identification</title>
# <summary> Recently, the theory of diffusion maps was extended to a large class of local
# kernels with exponential decay which were shown to represent various Riemannian
# geometries on a data set sampled from a manifold embedded in Euclidean space.
# Moreover, local kernels were used to represent a diffeomorphism, H, between a
# data set and a feature of interest using an anisotropic kernel function,
# defined by a covariance matrix based on the local derivatives, DH. In this
# paper, we generalize the theory of local kernels to represent degenerate
# mappings where the intrinsic dimension of the data set is higher than the
# intrinsic dimension of the feature space. First, we present a rigorous method
# with asymptotic error bounds for estimating DH from the training data set and
# feature values. We then derive scaling laws for the singular values of the
# local linear structure of the data, which allows the identification the tangent
# space and improved estimation of the intrinsic dimension of the manifold and
# the bandwidth parameter of the diffusion maps algorithm. Using these numerical
# tools, our approach to feature identification is to iterate the diffusion map
# with appropriately chosen local kernels that emphasize the features of
# interest. We interpret the iterated diffusion map (IDM) as a discrete
# approximation to an intrinsic geometric flow which smoothly changes the
# geometry of the data space to emphasize the feature of interest. When the data
# lies on a product manifold of the feature manifold with an irrelevant manifold,
# we show that the IDM converges to the quotient manifold which is isometric to
# the feature manifold, thereby eliminating the irrelevant dimensions. We will
# also demonstrate empirically that if we apply the IDM to features that are not
# a quotient of the data space, the algorithm identifies an intrinsically
# lower-dimensional set embedding of the data which better represents the
# features.
# </summary>
# <author>
# <name>Tyrus Berry</name>
# </author>
# <author>
# <name>John Harlim</name>
# </author>
# <link href="http://arxiv.org/abs/1509.07707v1" rel="alternate" type="text/html"/>
# <link title="pdf" href="http://arxiv.org/pdf/1509.07707v1" rel="related" type="application/pdf"/>
# <arxiv:primary_category xmlns:arxiv="http://arxiv.org/schemas/atom" term="math.CA" scheme="http://arxiv.org/schemas/atom"/>
# <category term="math.CA" scheme="http://arxiv.org/schemas/atom"/>
# </entry>
# <entry>
# <id>http://arxiv.org/abs/2006.09211v1</id>
# <updated>2020-06-16T14:43:24Z</updated>
# <published>2020-06-16T14:43:24Z</published>
# <title>A Note on the Axisymmetric Diffusion equation</title>
# <summary> We consider the explicit solution to the axisymmetric diffusion equation. We
# recast the solution in the form of a Mellin inversion formula, and outline a
# method to compute a formula for $u(r,t)$ as a series using the Cauchy residue
# theorem. As a consequence, we are able to represent the solution to the
# axisymmetric diffusion equation as rapidly converging series.
# </summary>
# <author>
# <name>Alexander E Patkowski</name>
# </author>
# <arxiv:doi xmlns:arxiv="http://arxiv.org/schemas/atom">10.1017/S1446181121000110</arxiv:doi>
# <link title="doi" href="http://dx.doi.org/10.1017/S1446181121000110" rel="related"/>
# <arxiv:journal_ref xmlns:arxiv="http://arxiv.org/schemas/atom">The ANZIAM Journal, Volume 63, Issue 3, July 2021, pp. 333--341</arxiv:journal_ref>
# <link href="http://arxiv.org/abs/2006.09211v1" rel="alternate" type="text/html"/>
# <link title="pdf" href="http://arxiv.org/pdf/2006.09211v1" rel="related" type="application/pdf"/>
# <arxiv:primary_category xmlns:arxiv="http://arxiv.org/schemas/atom" term="math.CA" scheme="http://arxiv.org/schemas/atom"/>
# <category term="math.CA" scheme="http://arxiv.org/schemas/atom"/>
# </entry>
# <entry>
# <id>http://arxiv.org/abs/1005.2786v1</id>
# <updated>2010-05-17T00:18:01Z</updated>
# <published>2010-05-17T00:18:01Z</published>
# <title>Positive travelling fronts for reaction-diffusion systems with
# distributed delay</title>
# <summary> We give sufficient conditions for the existence of positive travelling wave
# solutions for multi-dimensional autonomous reaction-diffusion systems with
# distributed delay. To prove the existence of travelling waves, we give an
# abstract formulation of the equation for the wave profiles in some suitable
# Banach spaces, and apply known results about the index of some associated
# Fredholm operators. After a Liapunov-Schmidt reduction, these waves are
# obtained via the Banach contraction principle, as perturbations of a positive
# heteroclinic solution for the associated system without diffusion, whose
# existence is proven under some requirements. By a careful analysis of the
# exponential decay of the travelling wave profiles at $-\infty$, their
# positiveness is deduced. The existence of positive travelling waves is
# important in terms of applications to biological models. Our method applies to
# systems of delayed reaction-diffusion equations whose nonlinearities are not
# required to satisfy a quasi-monotonicity condition. Applications are given, and
# include the delayed Fisher-KPP equation.
# </summary>
# <author>
# <name>Teresa Faria</name>
# </author>
# <author>
# <name>Sergei Trofimchuk</name>
# </author>
# <arxiv:doi xmlns:arxiv="http://arxiv.org/schemas/atom">10.1088/0951-7715/23/10/006</arxiv:doi>
# <link title="doi" href="http://dx.doi.org/10.1088/0951-7715/23/10/006" rel="related"/>
# <arxiv:comment xmlns:arxiv="http://arxiv.org/schemas/atom">28 pages, submitted</arxiv:comment>
# <arxiv:journal_ref xmlns:arxiv="http://arxiv.org/schemas/atom">Nonlinearity 23 (2010) 2457-2481</arxiv:journal_ref>
# <link href="http://arxiv.org/abs/1005.2786v1" rel="alternate" type="text/html"/>
# <link title="pdf" href="http://arxiv.org/pdf/1005.2786v1" rel="related" type="application/pdf"/>
# <arxiv:primary_category xmlns:arxiv="http://arxiv.org/schemas/atom" term="math.CA" scheme="http://arxiv.org/schemas/atom"/>
# <category term="math.CA" scheme="http://arxiv.org/schemas/atom"/>
# <category term="35K57, 35R10 (Primary), 92D25 (Secondary)" scheme="http://arxiv.org/schemas/atom"/>
# </entry>
# <entry>
# <id>http://arxiv.org/abs/1711.09634v1</id>
# <updated>2017-11-27T11:54:06Z</updated>
# <published>2017-11-27T11:54:06Z</published>
# <title>About the chemostat model with a lateral diffusive compartment</title>
# <summary> We consider the classical chemostat model with an additional compartment
# connected by pure diffusion, and analyze its asymptotic properties. We
# investigate conditions under which this spatial structure is beneficial for
# species survival and yield conversion, compared to single chemostat. Moreover
# we look for the best structure (volume repartition and diffusion rate) which
# minimizes the volume required to attain a desired yield conversion. The
# analysis reveals that configurations with a single tank connected by diffusion
# to the input stream can be the most efficient.
# </summary>
# <author>
# <name>María Crespo</name>
# </author>
# <author>
# <name>Alain Rapaport</name>
# </author>
# <link href="http://arxiv.org/abs/1711.09634v1" rel="alternate" type="text/html"/>
# <link title="pdf" href="http://arxiv.org/pdf/1711.09634v1" rel="related" type="application/pdf"/>
# <arxiv:primary_category xmlns:arxiv="http://arxiv.org/schemas/atom" term="math.CA" scheme="http://arxiv.org/schemas/atom"/>
# <category term="math.CA" scheme="http://arxiv.org/schemas/atom"/>
# </entry>
# <entry>
# <id>http://arxiv.org/abs/1911.03261v2</id>
# <updated>2020-08-13T16:31:32Z</updated>
# <published>2019-11-08T13:49:35Z</published>
# <title>Regularity of the solution to fractional diffusion, advection, reaction
# equations</title>
# <summary> In this report we investigate the regularity of the solution to the
# fractional diffusion, advection, reaction equation on a bounded domain in
# $\mathbb{R}^{1}$. The analysis is performed in the weighted Sobolev spaces,
# $H_{(a , b)}^{s}(\mathrm{I})$. Three different characterizations of $H_{(a ,
# b)}^{s}(\mathrm{I})$ are presented, together with needed embedding theorems for
# these spaces. The analysis shows that the regularity of the solution is bounded
# by the endpoint behavior of the solution, which is determined by the parameters
# $\alpha$ and $r$ defining the fractional diffusion operator. Additionally, the
# analysis shows that for a sufficiently smooth right hand side function, the
# regularity of the solution to fractional diffusion reaction equation is lower
# than that of the fractional diffusion equation. Also, the regularity of the
# solution to fractional diffusion advection reaction equation is two orders
# lower than that of the fractional diffusion reaction equation.
# </summary>
# <author>
# <name>V. J. Ervin</name>
# </author>
# <arxiv:comment xmlns:arxiv="http://arxiv.org/schemas/atom">40 pages</arxiv:comment>
# <link href="http://arxiv.org/abs/1911.03261v2" rel="alternate" type="text/html"/>
# <link title="pdf" href="http://arxiv.org/pdf/1911.03261v2" rel="related" type="application/pdf"/>
# <arxiv:primary_category xmlns:arxiv="http://arxiv.org/schemas/atom" term="math.CA" scheme="http://arxiv.org/schemas/atom"/>
# <category term="math.CA" scheme="http://arxiv.org/schemas/atom"/>
# <category term="35R11, 35B65, 46E35" scheme="http://arxiv.org/schemas/atom"/>
# </entry>
# <entry>
# <id>http://arxiv.org/abs/1805.09398v1</id>
# <updated>2018-05-23T19:38:13Z</updated>
# <published>2018-05-23T19:38:13Z</published>
# <title>On Fractional Diffusion-Advection-Reaction Equation In $\mathbb{R}$</title>
# <summary> We present an analysis of existence, uniqueness, and smoothness of the
# solution to a class of fractional ordinary differential equations posed on the
# whole real line that models a steady state behavior of a certain anomalous
# diffusion, advection, and reaction. The anomalous diffusion is modeled by the
# fractional Riemann-Liouville differential operators. The strong solution of the
# equation is sought in a Sobolev space defined by means of Fourier Transform.
# The key component of the analysis hinges on a characterization of this Sobolev
# space with the Riemann-Liouville derivatives that are understood in a weak
# sense. The existence, uniqueness, and smoothness of the solution is
# demonstrated with the assistance of several tools from functional and harmonic
# analyses.
# </summary>
# <author>
# <name>V. Ginting</name>
# </author>
# <author>
# <name>Y. Li</name>
# </author>
# <link href="http://arxiv.org/abs/1805.09398v1" rel="alternate" type="text/html"/>
# <link title="pdf" href="http://arxiv.org/pdf/1805.09398v1" rel="related" type="application/pdf"/>
# <arxiv:primary_category xmlns:arxiv="http://arxiv.org/schemas/atom" term="math.CA" scheme="http://arxiv.org/schemas/atom"/>
# <category term="math.CA" scheme="http://arxiv.org/schemas/atom"/>
# </entry>
# <entry>
# <id>http://arxiv.org/abs/0704.1916v2</id>
# <updated>2007-08-07T16:15:22Z</updated>
# <published>2007-04-15T19:20:52Z</published>
# <title>Solutions of certain fractional kinetic equations and a fractional
# diffusion equation</title>
# <summary> In view of the usefulness and importance of the kinetic equation in certain
# physical problems, the authors derive the explicit solution of a fractional
# kinetic equation of general character, that unifies and extends earlier
# results. Further, an alternative shorter method based on a result developed by
# the authors is given to derive the solution of a fractional diffusion equation.
# </summary>
# <author>
# <name>R. K. Saxena</name>
# </author>
# <author>
# <name>A. M. Mathai</name>
# </author>
# <author>
# <name>H. J. Haubold</name>
# </author>
# <arxiv:doi xmlns:arxiv="http://arxiv.org/schemas/atom">10.1063/1.3496829</arxiv:doi>
# <link title="doi" href="http://dx.doi.org/10.1063/1.3496829" rel="related"/>
# <arxiv:comment xmlns:arxiv="http://arxiv.org/schemas/atom">10 pages, LaTeX,corrected typos</arxiv:comment>
# <arxiv:journal_ref xmlns:arxiv="http://arxiv.org/schemas/atom">Journal of Mathematical Physics 51(2010)103506</arxiv:journal_ref>
# <link href="http://arxiv.org/abs/0704.1916v2" rel="alternate" type="text/html"/>
# <link title="pdf" href="http://arxiv.org/pdf/0704.1916v2" rel="related" type="application/pdf"/>
# <arxiv:primary_category xmlns:arxiv="http://arxiv.org/schemas/atom" term="math.CA" scheme="http://arxiv.org/schemas/atom"/>
# <category term="math.CA" scheme="http://arxiv.org/schemas/atom"/>
# <category term="33C20, 33C90" scheme="http://arxiv.org/schemas/atom"/>
# </entry>
# <entry>
# <id>http://arxiv.org/abs/0802.0463v1</id>
# <updated>2008-02-04T17:43:42Z</updated>
# <published>2008-02-04T17:43:42Z</published>
# <title>The multi-dimensional pencil phenomenon for Laguerre heat-diffusion
# maximal operators</title>
# <summary> We investigate in detail the mapping properties of the maximal operator
# associated with the heat-diffusion semigroup corresponding to expansions with
# respect to multi-dimensional standard Laguerre functions.
# </summary>
# <author>
# <name>Adam Nowak</name>
# </author>
# <author>
# <name>Peter Sjögren</name>
# </author>
# <arxiv:comment xmlns:arxiv="http://arxiv.org/schemas/atom">30 pages with 1 figure. This paper was published as Preprint 2007:35,
# Department of Mathematical Sciences, Chalmers University of Technology and
# University of Gothenburg</arxiv:comment>
# <arxiv:journal_ref xmlns:arxiv="http://arxiv.org/schemas/atom">Math. Ann. 344 (2009), 213-248.</arxiv:journal_ref>
# <link href="http://arxiv.org/abs/0802.0463v1" rel="alternate" type="text/html"/>
# <link title="pdf" href="http://arxiv.org/pdf/0802.0463v1" rel="related" type="application/pdf"/>
# <arxiv:primary_category xmlns:arxiv="http://arxiv.org/schemas/atom" term="math.CA" scheme="http://arxiv.org/schemas/atom"/>
# <category term="math.CA" scheme="http://arxiv.org/schemas/atom"/>
# <category term="42C10 (Primary); 42B25 (Secondary)" scheme="http://arxiv.org/schemas/atom"/>
# </entry>
# <entry>
# <id>http://arxiv.org/abs/1509.04789v1</id>
# <updated>2015-09-16T02:02:27Z</updated>
# <published>2015-09-16T02:02:27Z</published>
# <title>Monotone waves for non-monotone and non-local monostable
# reaction-diffusion equations</title>
# <summary> We propose a criterion for the existence of monotone wavefronts in
# non-monotone and non-local monostable diffusive equations of the Mackey-Glass
# type. This extends recent results by Gomez et al proved for the particular case
# of equations with local delayed reaction. In addition, we demonstrate the
# uniqueness (up to a translation) of obtained monotone wavefront within the
# class of all monotone wavefronts (such a kind of conditional uniqueness was
# recently established for the non-local KPP-Fisher equation by Fang and Zhao).
# Moreover, we show that if delayed reaction is local then this uniqueness
# actually holds within the class of all wavefronts and therefore the minimal
# fronts under consideration (either pulled or pushed) should be monotone.
# Similarly to the case of the KPP-Fisher equations, our approach is based on the
# construction of an appropriate fundamental solution for associated boundary
# value problem for linear integral-differential equation.
# </summary>
# <author>
# <name>Elena Trofimchuk</name>
# </author>
# <author>
# <name>Manuel Pinto</name>
# </author>
# <author>
# <name>Sergei Trofimchuk</name>
# </author>
# <arxiv:doi xmlns:arxiv="http://arxiv.org/schemas/atom">10.1016/j.jde.2016.03.039</arxiv:doi>
# <link title="doi" href="http://dx.doi.org/10.1016/j.jde.2016.03.039" rel="related"/>
# <arxiv:comment xmlns:arxiv="http://arxiv.org/schemas/atom">23 pages, submitted</arxiv:comment>
# <arxiv:journal_ref xmlns:arxiv="http://arxiv.org/schemas/atom">Journal of Differential Equations 261 (2016) pp. 1203-1236</arxiv:journal_ref>
# <link href="http://arxiv.org/abs/1509.04789v1" rel="alternate" type="text/html"/>
# <link title="pdf" href="http://arxiv.org/pdf/1509.04789v1" rel="related" type="application/pdf"/>
# <arxiv:primary_category xmlns:arxiv="http://arxiv.org/schemas/atom" term="math.CA" scheme="http://arxiv.org/schemas/atom"/>
# <category term="math.CA" scheme="http://arxiv.org/schemas/atom"/>
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