The NSF Research Experiences for Undergraduates Site in the UConn Math Department has been renewed. This grant supports approximately 9 undergraduates each year to come to UConn and do research with faculty and graduate student mentors.
Author: Rogers, Luke
Analysis, Probability and Mathematical Physics on Fractals
edited by Patricia Alonso Ruiz, Joe P Chen, Luke G Rogers, Robert S Strichartz and Alexander Teplyaev
In the 50 years since Mandelbrot identified the fractality of coastlines, mathematicians and physicists have developed a rich and beautiful theory describing the interplay between analytic, geometric and probabilistic aspects of the mathematics of fractals. Continue reading
Discretization of the Koch Snowflake Domain with Boundary and Interior Energies
by Malcolm Gabbard, Carlos Lima, Gamal Mograby, Luke G. Rogers, Alexander Teplyaev.
We study the discretization of a Dirichlet form on the Koch snowflake domain and its boundary with the property that both the interior and the boundary can support positive energy. Continue reading
Geodesic Interpolation on Sierpinski Gaskets
by Caitlin M. Davis, Laura A. LeGare, Cory W. McCartan, Luke G. Rogers
We study the analogue of a convex interpolant of two sets on Sierpinski gaskets and an associated notion of measure transport. Continue reading
BV functions and fractional Laplacians on Dirichlet spaces
by Patricia Alonso Ruiz, Fabrice Baudoin, Li Chen, Luke Rogers, Nageswari Shanmugalingam, Alexander Teplyaev
We study $$L^p$$ Besov critical exponents and isoperimetric and Sobolev inequalities associated with fractional Laplacians on metric measure spaces. Continue reading
Harmonic Gradients on Higher Dimensional Sierpinski Gaskets
by Luke Brown, Giovanni Ferrer, Gamal Mograby, Luke G. Rogers, Karuna Sangam
We consider criteria for the differentiability of functions with continuous Laplacian on the Sierpinski Gasket and its higher-dimensional variants $$SG_N$$, N>3, Continue reading
Spectral properties of graphs associated to the Basilica group
by Antoni Brzoska, Courtney George, Samantha Jarvis, Luke G. Rogers, Alexander Teplyaev
We provide the foundation of the spectral analysis of the Laplacian on the orbital Schreier graphs of the basilica group, the iterated monodromy group of the quadratic polynomial $$z^2−1$$. Continue reading
Besov class via heat semigroup on Dirichlet spaces I: Sobolev type inequalities
by Patricia Alonso-Ruiz, Fabrice Baudoin, Li Chen, Luke Rogers, Nageswari Shanmugalingam, Alexander Teplyaev
We introduce heat semigroup-based Besov classes in the general framework of Dirichlet spaces. General properties of those classes are studied and quantitative regularization estimates for the heat semigroup in this scale of spaces are obtained. Continue reading
Besov class via heat semigroup on Dirichlet spaces III: BV functions and sub-Gaussian heat kernel estimate
by Patricia Alonso-Ruiz, Fabrice Baudoin, Li Chen, Luke Rogers, Nageswari Shanmugalingam, Alexander Teplyaev
With a view toward fractal spaces, by using a Korevaar-Schoen space approach, we introduce the class of bounded variation (BV) functions in a general framework of strongly local Dirichlet spaces with a heat kernel satisfying sub-Gaussian estimates. Continue reading
The Strong Maximum Principle for Schrödinger operators on fractals
by Marius V. Ionescu, Kasso A. Okoudjou, Luke G. Rogers
We prove a strong maximum principle for Schrödinger operators defined on a class of fractal sets and their blowups without boundary. Our primary interest is in weaker regularity conditions than have previously appeared in the literature; in particular we permit both the fractal Laplacian and the potential to be Radon measures on the fractal. As a consequence of our results, we establish a Harnack inequality for solutions of these operators.
Fractal AC circuits and propagating waves on fractals
by Eric Akkermans, Joe P. Chen, Gerald Dunne, Luke G. Rogers and Alexander Teplyaev
We extend Feynman’s analysis of the infinite ladder AC circuit to fractal AC circuits. We show that the characteristic impedances can have positive real part even though all the individual impedances inside the circuit are purely imaginary. Continue reading
Besov class via heat semigroup on Dirichlet spaces II: BV functions and Gaussian heat kernel estimates
by Patricia Alonso-Ruiz, Fabrice Baudoin, Li Chen, Luke Rogers, Nageswari Shanmugalingam, Alexander Teplyaev
We introduce the class of bounded variation (BV) functions in a general framework of strictly local Dirichlet spaces with doubling measure. Continue reading
BV functions and Besov spaces associated with Dirichlet spaces
by Patricia Alonso-Ruiz, Fabrice Baudoin, Li Chen, Luke Rogers, Nageswari Shanmugalingam, Alexander Teplyaev
We develop a theory of bounded variation functions and Besov spaces in abstract Dirichlet spaces which unifies several known examples and applies to new situations, including fractals. Continue reading
Spectra of Magnetic Operators on the Diamond Lattice Fractal
by Antoni Brzoska, Aubrey Coffey, Madeline Hansalik, Stephen Loew, Luke G. Rogers
We adapt the well-known spectral decimation technique for computing spectra of Laplacians on certain symmetric self-similar sets to the case of magnetic Schrödinger operators and work through this method completely for the diamond lattice fractal. Continue reading
Embedding convex geometries and a bound on convex dimension
by Michael Richter, Luke G. Rogers
The notion of an abstract convex geometry offers an abstraction of the standard notion of convexity in a linear space. Continue reading
Sobolev Algebra Counterexamples
by Thierry Coulhon, Luke G. Rogers
In the Euclidean setting the Sobolev spaces $$W^{α,p}\cap L^\infty$$ are algebras for the pointwise product when α>0 and p∈(1,∞). This property has recently been extended to a variety of geometric settings. We produce a class of fractal examples where it fails for a wide range of the indices α,p.
Power dissipation in fractal AC circuits
by Joe P Chen, Luke G. Rogers, Loren Anderson, Ulysses Andrews, Antoni Brzoska, Aubrey Coffey, Hannah Davis, Lee Fisher, Madeline Hansalik, Stephen Loew and Alexander Teplyaev.
We extend Feynman’s analysis of an infinite ladder circuit to fractal circuits, providing examples in which fractal circuits constructed with purely imaginary impedances can have characteristic impedances with positive real part. Continue reading
Magnetic Laplacians of locally exact forms on the Sierpinski Gasket
by Jessica Hyde, Daniel Kelleher, Jesse Moeller, Luke G. Rogers, Luis Seda
We give an explicit construction of a magnetic Schrödinger operator corresponding to a field with flux through finitely many holes of the Sierpinski Gasket. Continue reading
Some spectral properties of pseudo-differential operators on the Sierpinski Gasket
by Marius Ionescu, Kasso A. Okoudjou and Luke G. Rogers.
We prove versions of the strong Szëgo limit theorem for certain classes of pseudodifferential operators defined on the Sierpinski gasket. Our results uses in a fundamental way the existence of localized eigenfunctions for the Laplacian on this fractal. Continue reading
On a theorem of Grigorʹyan, Hu and Lau
by Luke Rogers
We refine a result of Grigorʹyan, Hu and Lau to give a moment condition on a heat kernel which characterizes the critical exponent at which a family of Besov spaces associated to the Dirichlet energy becomes trivial. Continue reading
Pseudo-differential Operators on Fractals
by Marius Ionescu, Luke G. Rogers and Robert S. Strichartz
We define and study pseudo-differential operators on a class of fractals that include the post-critically finite self-similar sets and Sierpinski carpets. Continue reading
Derivations and Dirichlet forms on fractals
by Marius Ionescu, Luke G. Rogers and Alexander Teplyaev
We study derivations and Fredholm modules on metric spaces with a local regular conservative Dirichlet form. Continue reading
Detecting stable scales in images via non-smooth K-functionals
by Triet M. Le and Luke Rogers
In image decompositions, one is interested in decomposing $$f$$ into $$u+v$$ where $$u$$ and $$v$$ have different features. In a variational approach, such a decomposition is achieved by solving the following variational problem
$$ \inf_{(u,v)∈X1×X2} \{ tF_1(u) + F_2(v) : f = u + v\}$$, Continue reading
Estimates for the resolvent kernel of the Laplacian on p.c.f. self similar fractals and blowups
by Luke G. Rogers
We provide a method for obtaining upper estimates of the resolvent kernel of the Laplacian on a post-critically finite self-similar fractal that relies on a self-similar series decomposition of the resolvent. Continue reading
Complex Powers of the Laplacian on Affine Nested Fractals as Calderón-Zygmund operators
by Marius Ionescu and Luke G. Rogers
We give the first natural examples of Calderón-Zygmund operators in the theory of analysis on post-critically finite self-similar fractals. Continue reading
Smooth bumps, a Borel theorem and partitions of smooth functions on p.c.f. fractals.
by Luke G. Rogers, Robert S. Strichartz and Alexander Teplyaev
We provide two methods for constructing smooth bump functions and for smoothly cutting off smooth
functions on fractals, one using a probabilistic approach and sub-Gaussian estimates for the heat
operator, and the other using the analytic theory for p.c.f. fractals and a fixed point argument. Continue reading
The resolvent kernel for PCF self-similar fractals
by Marius Ionescu, Erin P. J. Pearse, Luke G. Rogers, Huo-Jun Ruan, Robert S. Strichartz
For the Laplacian Δ defined on a p.c.f. self-similar fractal, we give an explicit formula for the resolvent kernel of the Laplacian with Dirichlet boundary conditions, Continue reading
Distribution theory on p.c.f. fractals
by Luke G. Rogers, Robert S. Strichartz
We construct a theory of distributions in the setting of analysis on post-critically finite self-similar fractals, and on fractafolds and products based on such fractals. Continue reading
Szegö limit theorems on the Sierpinski gasket
by Kasso A. Okoudjou, Luke G. Rogers, Robert S. Strichartz
We use the existence of localized eigenfunctions of the Laplacian on the Sierpinski gasket to formulate and prove analogues of the strong Szego limit theorem in this fractal setting. Furthermore, we recast some of our results in terms of equally distributed sequences. Continue reading
Laplacians on the basilica Julia set
by Luke G Rogers and Alexander Teplyaev.
We consider the basilica Julia set of the polynomial $$P(z)=z^{2}-1$$ and construct all possible
resistance (Dirichlet) forms, and the corresponding Laplacians, for which the topology in the
effective resistance metric coincides with the usual topology. Continue reading
Gradients of Laplacian Eigenfunctions on the Sierpinski Gasket
by Jessica L. DeGrado, Luke G. Rogers, Robert S. Strichartz
We use spectral decimation to provide formulae for computing the harmonic gradients of Laplacian eigenfunctions on the Sierpinski Gasket. These formulae are given in terms of special functions that are defined as infinite products. Continue reading
Unimodular Multipliers on Modulation Spaces.
by Árpád Bényi, Kasso A. Okoudjou, Karlheniz Gröchenig and Luke G. Rogers.
We investigate the boundedness of unimodular Fourier multipliers on modulation spaces. Surprisingly, the multipliers with general symbol $$e^{i|\xi|^\alpha}$$ where $$\alpha\in[0,2]$$, are bounded on all modulation spaces, Continue reading
Generalized Eigenfunctions and a Borel Theorem on the Sierpinski Gasket
by Kasso A Okoudjou, Luke G. Rogers and Robert S. Strichartz
We prove there exist exponentially decaying generalized eigenfunctions on a blow-up of the Sierpinski gasket with boundary. These are used to show a Borel-type theorem, specifically that for a prescribed jet at the boundary point there is a smooth function having that jet. Continue reading
Degree-independent Sobolev extension on locally uniform domains
by Luke G. Rogers
We consider the problem of constructing extensions $$L^{p}_{k}(\Omega)\rightarrow L^{p}_{k}(\mathbb{R}^{n})$$, where $$L^{p}_{k}$$ is the Sobolev space of functions with $$k$$ derivatives in $$L^{p}$$ and $$\Omega\subset\mathbb{R}^{n}$$ is a domain. Continue reading