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David Larson

Publications and source records attributed to David Larson.

At least 19 recordsLinked to original sources
arXiv2025

Dynamical Frames and Hyperinvariant Subspaces

The theory of dynamical frames evolved from practical problems in dynamical sampling where the initial state of a vector needs to be recovered from the space-time samples of evolutions of the vector. This leads to the investigation of structured frames obtained from the orbits of evolution operators. One of the basic problems in dynamical frame theory is to determine the semigroup representations, which we will call central frame representations, whose frame generators are unique (up to equivalence). Recently, Christensen, Hasannasab, and Philipp proved that all frame representations of the semigroup $\Bbb{Z}_{+}$ have this property. Their proof of this result relies on the characterization of the structure of shift-invariant subspaces in $H^2(\mathbb{D})$ due to Beurling. In this paper we settle the general uniqueness problem by presenting a characterization of central frame representations for any semigroup in terms of the co-hyperinvariant subspaces of the left regular representation of the semigroup. This result is not only consistent with the known result of Han-Larson in 2000 for group representation frames, but also proves that all the frame generators of a semigroup generated by any $k$-tuple $(A_1, ... A_k)$ of commuting bounded linear operators on a separable Hilbert space $H$ are equivalent, a case where the structure of shift-invariant subspaces, or submodules, of the Hardy Space on polydisks $H^{2}(\Bbb{D}^k)$ is still not completely characterized.

math.FA↗
arXiv2024

Moment-preserving Monte-Carlo Coulomb collision method for particle codes

Binary-pairing Monte-Carlo methods are widely used in particle-in-cell codes to capture effects of small angle Coulomb collisions. These methods preserve momentum and energy exactly when the simulation particles have equal weights. However, when the interacting particles are of varying weight, these physical conservation laws are only preserved on average. Here, we 1) extend these methods to weighted particles such that the scattering physics is correct on average, and 2) describe a new method for adjusting the particle velocities post scatter to restore exact conservation of momentum and energy. The efficacy of the model is illustrated with various test problems.

physics.comp-ph↗
arXiv2023

A nanoporous capacitive electrochemical ratchet for continuous ion separations

Directed ion transport in liquid electrolyte solutions underlies many phenomena in Nature and industry. While Nature has devised structures that drive continuous ion flow without Faradaic redox reactions, artificial analogs do not exist. Here we report the first demonstration of an ion pump that drives aqueous ions against a force using a capacitive ratchet mechanism that does not require redox reactions. Modulation of an electric potential between thin metallic layers on either face of a nanoporous alumina wafer immersed in solution resulted in persistent voltages and ionic currents. This occurs due to the non-linear capacitive nature of electric double layers, whose repeated charging and discharging sustains a continuous ion flux. Ratchet driven electrodialysis was demonstrated reaching a 50% decrease in the conductivity of the solution in a dilution cell. These ratchet-based ion pumps can enable continuous desalination and selective ion separation using an electrically powered device with no moving parts.

physics.app-ph↗
arXiv2020

Development of a low flow vapor phase electrochemical reactor without catholyte reliance for CO2 electrolysis to high value carbon products

A continuous CO2 vapor-fed electrochemical cell prototype that performs CO2R with high faradaic efficiency for desired carbon products for up to 72 hours of operation is presented. The cell design facilitates a flow through configuration, capitalizing on gas diffusion electrode (GDE) and membrane electrode architecture (MEA) without requiring a catholyte. We demonstrate stable performance and design adaptability by incorporating various CO2R catalysts and membranes into the cell, investigating lifetime performance and selectivity under established experimental conditions. With an Ag foil and PPO anion exchange membrane, the design achieves an average current density of -30mA/cm2 and FE ~80% for CO obtained over a 12-hour duration. With a Cu-based catalyst, FE ~40% selectivity for ethylene was achieved. The unique geometry and flexibility of the cell provides an adaptive and dynamic GDE electrochemical cell framework, easing future research for novel electro-catalysts on GDE electrodes and MEA/GDE assemblies for enhanced (photo)electrochemical carbon dioxide reduction.

cond-mat.mtrl-sci↗
arXiv2018

A duality principle for groups II: Multi-frames meet super-frames

The duality principle for group representations developed in \cite{DHL-JFA, HL_BLM} exhibits a fact that the well-known duality principle in Gabor analysis is not an isolated incident but a more general phenomenon residing in the context of group representation theory. There are two other well-known fundamental properties in Gabor analysis: The Wexler-Raz biorthogonality and the Fundamental Identity of Gabor analysis. In this paper we will show that these fundamental properties remain to be true for general projective unitary group representations. The main purpose of this paper is present a more general duality theorem which shows that that muti-frame generators meet super-frame generators through a dual commutant pairs. In particular, for the Gabor representations $\pi_{\Lambda}$ and $\pi_{\Lambda^{o}}$ with respect to a pair of dual time-frequency lattices $\Lambda$ and $\Lambda^{o}$ in $\R^{d}\times \R^{d}$ we have that $\{\pi_{\Lambda}(m, n)g_{1} \oplus ... \oplus \pi_{\Lambda}(m, n)g_{k}\}_{m, n \in \Z^{d}}$ is a frame for $L^{2}(\R^{d})\oplus ... \oplus L^{2}(\R^{d})$ if and only if $\cup_{i=1}^{k}\{\pi_{\Lambda^{o}}(m, n)g_{i}\}_{m, n\in\Z^{d}}$ is a Riesz sequence, and $\cup_{i=1}^{k}\{\pi_{\Lambda}(m, n)g_{i}\}_{m, n\in\Z^{d}}$ is a frame for $L^{2}(\R^{d})$ if and only if $\{\pi_{\Lambda^{o}}(m, n)g_{1} \oplus ... \oplus \pi_{\Lambda^{o}}(m, n)g_{k}\}_{m, n \in \Z^{d}}$ is a Riesz sequence. This appears to be new even in the context of Gabor analysis.

math.FA↗
arXiv2018

Admissible sequences for positive operators

A sequence of scalars is said to be admissible for a positive operator A on a Hilbert space if it is the diagonal of VAV* for some partial isometry V having as domain the closure of the range of A. When A is a projection, the celebrated Kadison's carpenter theorem provides a sufficient condition for a sequence to be admissible for A. We prove that the same condition is sufficient for the sequence to be admissible for A when A is a sum of projections (converging in the SOT). This provides an independent proof of Kadison's carpenter theorem.

math.OA↗
arXiv2015

Measuring the Largest Angular Scale CMB B-mode Polarization with Galactic Foregrounds on a Cut Sky

We consider the effectiveness of foreground cleaning in the recovery of Cosmic Microwave Background (CMB) polarization sourced by gravitational waves for tensor-to-scalar ratios in the range $0<r<0.1$. Using the planned survey area, frequency bands, and sensitivity of the Cosmology Large Angular Scale Surveyor (CLASS), we simulate maps of Stokes $Q$ and $U$ parameters at 40, 90, 150, and 220 GHz, including realistic models of the CMB, diffuse Galactic thermal dust and synchrotron foregrounds, and Gaussian white noise. We use linear combinations (LCs) of the simulated multifrequency data to obtain maximum likelihood estimates of $r$, the relative scalar amplitude $s$, and LC coefficients. We find that for 10,000 simulations of a CLASS-like experiment using only measurements of the reionization peak ($\ell\leq23$), there is a 95% C.L. upper limit of $r<0.017$ in the case of no primordial gravitational waves. For simulations with $r=0.01$, we recover at 68% C.L. $r=0.012^{+0.011}_{-0.006}$. The reionization peak corresponds to a fraction of the multipole moments probed by CLASS, and simulations including $30\leq\ell\leq100$ further improve our upper limits to $r<0.008$ at 95% C.L. ($r=0.01^{+0.004}_{-0.004}$ for primordial gravitational waves with $r=0.01$). In addition to decreasing the current upper bound on $r$ by an order of magnitude, these foreground-cleaned low multipole data will achieve a cosmic variance limited measurement of the E-mode polarization's reionization peak.

astro-ph.CO↗
arXiv2014

The Cosmology Large Angular Scale Surveyor (CLASS): 38 GHz detector array of bolometric polarimeters

The Cosmology Large Angular Scale Surveyor (CLASS) experiment aims to map the polarization of the Cosmic Microwave Background (CMB) at angular scales larger than a few degrees. Operating from Cerro Toco in the Atacama Desert of Chile, it will observe over 65% of the sky at 38, 93, 148, and 217 GHz. In this paper we discuss the design, construction, and characterization of the CLASS 38 GHz detector focal plane, the first ever Q-band bolometric polarimeter array.

astro-ph.IM↗
arXiv2014

CLASS: The Cosmology Large Angular Scale Surveyor

The Cosmology Large Angular Scale Surveyor (CLASS) is an experiment to measure the signature of a gravita-tional-wave background from inflation in the polarization of the cosmic microwave background (CMB). CLASS is a multi-frequency array of four telescopes operating from a high-altitude site in the Atacama Desert in Chile. CLASS will survey 70\% of the sky in four frequency bands centered at 38, 93, 148, and 217 GHz, which are chosen to straddle the Galactic-foreground minimum while avoiding strong atmospheric emission lines. This broad frequency coverage ensures that CLASS can distinguish Galactic emission from the CMB. The sky fraction of the CLASS survey will allow the full shape of the primordial B-mode power spectrum to be characterized, including the signal from reionization at low $\ell$. Its unique combination of large sky coverage, control of systematic errors, and high sensitivity will allow CLASS to measure or place upper limits on the tensor-to-scalar ratio at a level of $r=0.01$ and make a cosmic-variance-limited measurement of the optical depth to the surface of last scattering, $\tau$.

astro-ph.IM↗
arXiv2009

A Duality principle for groups

The duality principle for Gabor frames states that a Gabor sequence obtained by a time-frequency lattice is a frame for $L^{2}(\R^{d})$ if and only if the associated adjoint Gabor sequence is a Riesz sequence. We prove that this duality principle extends to any dual pairs of projective unitary representations of countable groups. We examine the existence problem of dual pairs and establish some connection with classification problems for $\rm II_{1}$ factors. While in general such a pair may not exist for some groups, we show that such a dual pair always exists for every subrepresentation of the left regular unitary representation when $G$ is an abelian infinite countable group or an amenable ICC group. For free groups with finitely many generators, the existence problem of such a dual pair is equivalent to the well known problem about the classification of free group von Neumann algebras.

math.FA↗
arXiv2007

Coxeter Groups and Wavelet Sets

A traditional wavelet is a special case of a vector in a separable Hilbert space that generates a basis under the action of a system of unitary operators defined in terms of translation and dilation operations. A Coxeter/fractal-surface wavelet is obtained by defining fractal surfaces on foldable figures, which tesselate the embedding space by reflections in their bounding hyperplanes instead of by translations along a lattice. Although both theories look different at their onset, there exist connections and communalities which are exhibited in this semi-expository paper. In particular, there is a natural notion of a dilation-reflection wavelet set. We prove that dilation-reflection wavelet sets exist for arbitrary expansive matrix dilations, paralleling the traditional dilation-translation wavelet theory. There are certain measurable sets which can serve simultaneously as dilation-translation wavelet sets and dilation-reflection wavelet sets, although the orthonormal structures generated in the two theories are considerably different.

math.FA↗
arXiv2007

Three-way tiling sets in two dimensions

In this article we show that there exist measurable sets W in the plane with finite measure that tile the plane in a measurable way under the action of a expansive matrix A, an affine Weyl group W, and a full rank lattice G. This note is follow-up research to the earlier article "Coxeter groups and wavelet sets" by the first and second authors, and is also relevant to the earlier article "Coxeter groups, wavelets, multiresolution and sampling" by M. Dobrescu and the third author. After writing these two articles, the three authors participated in a workshop at the Banff Center on "Operator methods in fractal analysis, wavelets and dynamical systems," December 2 -- 7, 2006, organized by O. Bratteli, P. Jorgensen, D. Kribs, G. Olafsson, and S. Silvestrov, and discussed the interrelationships and differences between the articles, and worked on two open problems posed in the Larson-Massopust article. We solved part of Problem 2, including a surprising positive solution to a conjecture that was raised, and we present our results in this article.

math.FA↗
arXiv2007

Operator valued frames on C*-modules

Frames on Hilbert C*-modules have been defined for unital C*-algebras by Frank and Larson and operator valued frames on a Hilbert space have been studied in arXiv.0707.3272v1.[math.FA]. Goal of the present paper is to introduce operator valued frames on a Hilbert C*-module for a sigma-unital C*-algebra. Theorem 1.4 reformulates the definition given by Frank and Larson in terms of a series of rank-one operators converging in the strict topology. Theorem 2.2. shows that the frame transform and the frame projection of an operator valued frame are limits in the strict topology of a series of elements in the multiplier algebra and hence belong to it. Theorem 3.3 shows that two operator valued frames are right similar if and only if they share the same frame projection. Theorem 3.4 establishes a one to one correspondence between Murray-von Neumann equivalence classes of projections in the multiplier algebra and right similarity equivalence classes of operator valued frames and provides a parametrization of all Parseval operator-valued frames on a given Hilbert C*-module. Left similarity is then defined and Proposition 3.9 establishes when two left unitarily equivalent frames are also right unitarily equivalent.

math.OA↗
arXiv2007

Operator valued frames

We develop a natural generalization of vector-valued frame theory, we term operator-valued frame theory, using operator-algebraic methods. This extends work of the second author and D. Han which can be viewed as the multiplicity one case and extends to higher multiplicity (e.g., multiframes) their dilation approach. We prove several results for operator-valued frames concerning their parametrization, duality, disjointeness, complementarity, and composition and the relationship between the two types of similarity (left and right) of such frames. We then apply these notions to prove that the collection of multiframe generators for the action of a discrete group on a Hilbert space is norm pathwise-connected precisely when the von Neumann algebra generated by the right representation of the group has no minimal projections. The proof is obtained by parametrizing this collection by a class of partial isometries in a larger von Neumann algebra. In the multiplicity one case this class reduces to the unitary class which is path-connected in norm, but in the infinite multiplicity case this class is path connected only in the strong operator topology and the proof depends on properties of tensor product slice maps.

math.FA↗
arXiv2006

Explicit cross-sections of singly generated group actions

We consider two classes of actions on $\mathbb{R}^n$ - one continuous and one discrete. For matrices of the form $A = e^B$ with $B \in M_n(\R)$, we consider the action given by $γ\to γA^t$. We characterize the matrices $A$ for which there is a cross-section for this action. The discrete action we consider is given by $γ\to γA^k$, where $A\in GL_n(\R)$. We characterize the matrices $A$ for which there exists a cross-section for this action as well. We also characterize those $A$ for which there exist special types of cross-sections; namely, bounded cross-sections and finite measure cross-sections. Explicit examples of cross-sections are provided for each of the cases in which cross-sections exist. Finally, these explicit cross-sections are used to characterize those matrices for which there exist MSF wavelets with infinitely many wavelet functions. Along the way, we generalize a well-known aspect of the theory of shift-invariant spaces to shift-invariant spaces with infinitely many generators.

math.FA↗
arXiv2003

Rank-One Decomposition of Operators and Construction of Frames

The construction of frames for a Hilbert space H can be equated to the decomposition of the frame operator as a sum of positive operators having rank one. This realization provides a different approach to questions regarding frames with particular properties and motivates our results. We find a necessary and sufficient condition under which any positive finite-rank operator B can be expressed as a sum of rank-one operators with norms specified by a sequence of positive numbers {c_i}. Equivalently, this result proves the existence of a frame with B as it's frame operator and with vector norms given by the square roots of the sequence elements. We further prove that, given a non-compact positive operator B on an infinite dimensional separable real or complex Hilbert space, and given an infinite sequence {c_i} of positive real numbers which has infinite sum and which has supremum strictly less than the essential norm of B, there is a sequence of rank-one positive operators, with norms given by {c_i}, which sum to B in the strong operator topology. These results generalize results by Casazza, Kovacevic, Leon, and Tremain, in which the operator is a scalar multiple of the identity operator (or equivalently the frame is a tight frame), and also results by Dykema, Freeman, Kornelson, Larson, Ordower, and Weber in which {c_i} is a constant sequence.

math.FA↗
arXiv2003

Geometric Aspects of Frame Representations of Abelian Groups

We consider frames arising from the action of a unitary representation of a discrete countable abelian group. We show that the range of the analysis operator can be determined by computing which characters appear in the representation. This allows one to compare the ranges of two such frames, which is useful for determining similarity and also for multiplexing schemes. Our results then partially extend to Bessel sequences arising from the action of the group. We apply the results to sampling on bandlimited functions and to wavelet and Weyl-Heisenberg frames. This yields a sufficient condition for two sampling transforms to have orthogonal ranges, and two analysis operators for wavelet and Weyl-Heisenberg frames to have orthogonal ranges. The sufficient condition is easy to compute in terms of the periodization of the Fourier transform of the frame generators.

math.FA↗