4,333 research outputs found
An Adaptive Fast Solver for a General Class of Positive Definite Matrices Via Energy Decomposition
In this paper, we propose an adaptive fast solver for a general class of symmetric positive definite (SPD) matrices which include the well-known graph Laplacian. We achieve this by developing an adaptive operator compression scheme and a multiresolution matrix factorization algorithm which achieve nearly optimal performance on both complexity and well-posedness. To develop our adaptive operator compression and multiresolution matrix factorization methods, we first introduce a novel notion of energy decomposition for SPD matrix using the representation of energy elements. The interaction between these energy elements depicts the underlying topological structure of the operator. This concept of decomposition naturally reflects the hidden geometric structure of the operator which inherits the localities of the structure. By utilizing the intrinsic geometric information under this energy framework, we propose a systematic operator compression scheme for the inverse operator . In particular, with an appropriate partition of the underlying geometric structure, we can construct localized basis by using the concept of interior and closed energy. Meanwhile, two important localized quantities are introduced, namely, the error factor and the condition factor. Our error analysis results show that these two factors will be the guidelines for finding the appropriate partition of the basis functions such that prescribed compression error and acceptable condition number can be achieved. By virtue of this insight, we propose the patch pairing algorithm to realize our energy partition framework for operator compression with controllable compression error and condition number
A Fast Hierarchically Preconditioned Eigensolver Based on Multiresolution Matrix Decomposition
In this paper we propose a new iterative method to hierarchically compute a relatively large number of leftmost eigenpairs of a sparse symmetric positive matrix under the multiresolution operator compression framework. We exploit the well-conditioned property of every decomposition component by integrating the multiresolution framework into the implicitly restarted Lanczos method. We achieve this combination by proposing an extension-refinement iterative scheme, in which the intrinsic idea is to decompose the target spectrum into several segments such that the corresponding eigenproblem in each segment is well-conditioned. Theoretical analysis and numerical illustration are also reported to illustrate the efficiency and effectiveness of this algorithm
Combinatorial Hopf algebras and Towers of Algebras
Bergeron and Li have introduced a set of axioms which guarantee that the
Grothendieck groups of a tower of algebras can be
endowed with the structure of graded dual Hopf algebras. Hivert and Nzeutzhap,
and independently Lam and Shimozono constructed dual graded graphs from
primitive elements in Hopf algebras. In this paper we apply the composition of
these constructions to towers of algebras. We show that if a tower
gives rise to graded dual Hopf algebras then we must
have where .Comment: 7 page
An Adaptive Fast Solver for a General Class of Positive Definite Matrices Via Energy Decomposition
In this paper, we propose an adaptive fast solver for a general class of symmetric positive definite (SPD) matrices which include the well-known graph Laplacian. We achieve this by developing an adaptive operator compression scheme and a multiresolution matrix factorization algorithm which achieve nearly optimal performance on both complexity and well-posedness. To develop our adaptive operator compression and multiresolution matrix factorization methods, we first introduce a novel notion of energy decomposition for SPD matrix using the representation of energy elements. The interaction between these energy elements depicts the underlying topological structure of the operator. This concept of decomposition naturally reflects the hidden geometric structure of the operator which inherits the localities of the structure. By utilizing the intrinsic geometric information under this energy framework, we propose a systematic operator compression scheme for the inverse operator . In particular, with an appropriate partition of the underlying geometric structure, we can construct localized basis by using the concept of interior and closed energy. Meanwhile, two important localized quantities are introduced, namely, the error factor and the condition factor. Our error analysis results show that these two factors will be the guidelines for finding the appropriate partition of the basis functions such that prescribed compression error and acceptable condition number can be achieved. By virtue of this insight, we propose the patch pairing algorithm to realize our energy partition framework for operator compression with controllable compression error and condition number
A Fast Hierarchically Preconditioned Eigensolver Based on Multiresolution Matrix Decomposition
In this paper we propose a new iterative method to hierarchically compute a relatively large number of leftmost eigenpairs of a sparse symmetric positive matrix under the multiresolution operator compression framework. We exploit the well-conditioned property of every decomposition component by integrating the multiresolution framework into the implicitly restarted Lanczos method. We achieve this combination by proposing an extension-refinement iterative scheme, in which the intrinsic idea is to decompose the target spectrum into several segments such that the corresponding eigenproblem in each segment is well-conditioned. Theoretical analysis and numerical illustration are also reported to illustrate the efficiency and effectiveness of this algorithm
Intraspecfic variation in cold-temperature metabolic phenotypes of Arabidopsis lyrata ssp petraea
Atmospheric temperature is a key factor in determining the distribution of a plant species. Alongside this, plant populations growing at the margin of their range may exhibit traits that indicate genetic differentiation and adaptation to their local abiotic environment. We investigated whether geographically separated marginal populations of Arabidopsis lyrata ssp. petraea have distinct metabolic phenotypes associated with exposure to cold temperatures. Seeds of A. petraea were obtained from populations along a latitudinal gradient, namely Wales, Sweden and Iceland and grown in a controlled cabinet environment. Mannose, glucose, fructose, sucrose and raffinose concentrations were different between cold treatments and populations, especially in the Welsh population, but polyhydric alcohol concentrations were not. The free amino acid compositions were population specific, with fold differences in most amino acids, especially in the Icelandic populations, with gross changes in amino acids, particularly those associated with glutamine metabolism. Metabolic fingerprints and profiles were obtained. Principal component analysis (PCA) of metabolite fingerprints revealed metabolic characteristic phenotypes for each population and temperature. It is suggested that amino acids and carbohydrates were responsible for discriminating populations within the PCA. Metabolite fingerprinting and profiling has proved to be sufficiently sensitive to identify metabolic differences between plant populations at different atmospheric temperatures. These findings show that there is significant natural variation in cold metabolism among populations of A. l. petraea which may signify plant adaptation to local climates
Vortex lines in the three-dimensional XY model with random phase shifts
The stability of the ordered phase of the three-dimensional XY-model with
random phase shifts is studied by considering the roughening of a single
stretched vortex line due to the disorder. It is shown that the vortex line may
be described by a directed polymer Hamiltonian with an effective random
potential that is long range correlated. A Flory argument estimates the
roughness exponent to and the energy fluctuation exponent to
, thus fulfilling the scaling relation . The
Schwartz-Edwards method as well as a numerical integration of the corresponding
Burger's equation confirm this result. Since the ordered phase of the
original XY-model is stable.Comment: 8 pages RevTeX, 3 eps-figures include
Two-species percolation and Scaling theory of the metal-insulator transition in two dimensions
Recently, a simple non-interacting-electron model, combining local quantum
tunneling via quantum point contacts and global classical percolation, has been
introduced in order to describe the observed ``metal-insulator transition'' in
two dimensions [1]. Here, based upon that model, a two-species-percolation
scaling theory is introduced and compared to the experimental data. The two
species in this model are, on one hand, the ``metallic'' point contacts, whose
critical energy lies below the Fermi energy, and on the other hand, the
insulating quantum point contacts. It is shown that many features of the
experiments, such as the exponential dependence of the resistance on
temperature on the metallic side, the linear dependence of the exponent on
density, the scale of the critical resistance, the quenching of the
metallic phase by a parallel magnetic field and the non-monotonic dependence of
the critical density on a perpendicular magnetic field, can be naturally
explained by the model.
Moreover, details such as the nonmonotonic dependence of the resistance on
temperature or the inflection point of the resistance vs. parallel magnetic are
also a natural consequence of the theory. The calculated parallel field
dependence of the critical density agrees excellently with experiments, and is
used to deduce an experimental value of the confining energy in the vertical
direction. It is also shown that the resistance on the ``metallic'' side can
decrease with decreasing temperature by an arbitrary factor in the degenerate
regime ().Comment: 8 pages, 8 figure
Percolation-type description of the metal-insulator transition in two dimensions
A simple non-interacting-electron model, combining local quantum tunneling
and global classical percolation (due to a finite dephasing time at low
temperatures), is introduced to describe a metal-insulator transition in two
dimensions. It is shown that many features of the experiments, such as the
exponential dependence of the resistance on temperature on the metallic side,
the linear dependence of the exponent on density, the scale of the
critical resistance, the quenching of the metallic phase by a parallel magnetic
field and the non-monotonic dependence of the critical density on a
perpendicular magnetic field, can be naturally explained by the model.Comment: 4 pages, 4 figure
Dual Vortex Theory of Strongly Interacting Electrons: Non-Fermi Liquid to the (Hard) Core
As discovered in the quantum Hall effect, a very effective way for
strongly-repulsive electrons to minimize their potential energy is to aquire
non-zero relative angular momentum. We pursue this mechanism for interacting
two-dimensional electrons in zero magnetic field, by employing a representation
of the electrons as composite bosons interacting with a Chern-Simons gauge
field. This enables us to construct a dual description in which the fundamental
constituents are vortices in the auxiliary boson fields. The resulting
formalism embraces a cornucopia of possible phases. Remarkably,
superconductivity is a generic feature, while the Fermi liquid is not --
prompting us to conjecture that such a state may not be possible when the
interactions are sufficiently strong. Many aspects of our earlier discussions
of the nodal liquid and spin-charge separation find surprising incarnations in
this new framework.Comment: Modified dicussion of the hard-core model, correcting several
mistake
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