An official website of the United States government
Abstract We present a new decomposition of a Cauchy–Vandermonde matrix as a product of bidiagonal matrices which, unlike its existing bidiagonal decompositions, is now valid for a matrix of any rank. The new decompositions are insusceptible to the phenomenon known as subtractive cancellation in floating point arithmetic and are thus computable to high relative accuracy. In turn, other accurate matrix computations are also possible with these matrices, such as eigenvalue computation amongst others.
more »
« less
Fejér–Riesz factorization in the QRC-subalgebra and circularity of the quaternionic numerical range
Abstract We provide a characterization when the quaternionic numerical range of a matrix is a closed ball with center 0. The proof makes use of Fejér–Riesz factorization of matrix-valued trigonometric polynomials within the algebra of complex matrices associated with quaternion matrices.
more »
« less
- Award ID(s):
- 2000037
- PAR ID:
- 10512562
- Publisher / Repository:
- Springer Link
- Date Published:
- Journal Name:
- Advances in Operator Theory
- Volume:
- 9
- Issue:
- 2
- ISSN:
- 2662-2009
- Page Range / eLocation ID:
- 33
- Subject(s) / Keyword(s):
- Fejér–Riesz factorization Quaternionic numerical range Circularity QRC-subalgebra
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
Abstract Structural and mechanical cues from the extracellular matrix (ECM) regulate tissue morphogenesis. Tissue development has conventionally been studied withex vivosystems where mechanical properties of the extracellular environment are either poorly controlled in space and time, lack tunability, or do not mimic ECM mechanics. For these reasons, it remains unknown how matrix stress relaxation rate, a time-dependent mechanical property that influences several cellular processes, regulates mammary branching morphogenesis. Here, we systematically investigated the influence of matrix stress relaxation on mammary branching morphogenesis using 3D alginate-collagen matrices and spheroids of human mammary epithelial cells. Slow stress relaxing matrices promoted significantly greater branch formation compared to fast stress relaxing matrices. Branching in slow stress relaxing matrices was accompanied by local collagen fiber alignment, while collagen fibers remained randomly oriented in fast stress relaxing matrices. In slow stress relaxing matrices, branch formation was driven by intermittent pulling contractions applied to the local ECM at the tips of elongating branches, which was accompanied by an abundance of phosphorylated focal adhesion kinase (phospho-FAK) and β1 integrin at the tips of branches. On the contrary, we observed that growing spheroids in fast stress relaxing matrices applied isotropic pushing forces to the ECM. Pharmacological inhibition of both Rac1 and non-muscle myosin II prevented epithelial branch formation, regardless of matrix stress relaxation rate. Interestingly, restricting cellular expansion via increased osmotic pressure was sufficient to impede epithelial branching in slow stress relaxing matrices. This work highlights the importance of stress relaxation in regulating and directing mammary branch elongation.more » « less
-
Abstract Profile mixture models capture distinct biochemical constraints on the amino acid substitution process at different sites in proteins. These models feature a mixture of time-reversible models with a common matrix of exchangeabilities and distinct sets of equilibrium amino acid frequencies known as profiles. Combining the exchangeability matrix with each profile generates the matrix of instantaneous rates of amino acid exchange for that profile. Currently, empirically estimated exchangeability matrices (e.g. the LG matrix) are widely used for phylogenetic inference under profile mixture models. However, these were estimated using a single profile and are unlikely optimal for profile mixture models. Here, we describe the GTRpmix model that allows maximum likelihood estimation of a common exchangeability matrix under any profile mixture model. We show that exchangeability matrices estimated under profile mixture models differ from the LG matrix, dramatically improving model fit and topological estimation accuracy for empirical test cases. Because the GTRpmix model is computationally expensive, we provide two exchangeability matrices estimated from large concatenated phylogenomic-supermatrices to be used for phylogenetic analyses. One, called Eukaryotic Linked Mixture (ELM), is designed for phylogenetic analysis of proteins encoded by nuclear genomes of eukaryotes, and the other, Eukaryotic and Archaeal Linked mixture (EAL), for reconstructing relationships between eukaryotes and Archaea. These matrices, combined with profile mixture models, fit data better and have improved topology estimation relative to the LG matrix combined with the same mixture models. Starting with version 2.3.1, IQ-TREE2 allows users to estimate linked exchangeabilities (i.e. amino acid exchange rates) under profile mixture models.more » « less
-
The GraphBLAS standard (GraphBlas.org) is being developed to bring the potential of matrix-based graph algorithms to the broadest possible audience. Mathematically, the GraphBLAS defines a core set of matrix-based graph operations that can be used to implement a wide class of graph algorithms in a wide range of programming environments. This paper provides an introduction to the mathematics of the GraphBLAS. Graphs represent connections between vertices with edges. Matrices can represent a wide range of graphs using adjacency matrices or incidence matrices. Adjacency matrices are often easier to analyze while incidence matrices are often better for representing data. Fortunately, the two are easily connected by matrix multiplication. A key feature of matrix mathematics is that a very small number of matrix operations can be used to manipulate a very wide range of graphs. This composability of a small number of operations is the foundation of the GraphBLAS. A standard such as the GraphBLAS can only be effective if it has low performance overhead. Performance measurements of prototype GraphBLAS implementations indicate that the overhead is low.more » « less
-
Abstract This paper explores a novel revision of the Faddeev equation for three-body (3B) bound states, as initially proposed in Ref. [J. Golak, K. Topolnicki, R. Skibiński, W. Glöckle, H. Kamada, A. Nogga, Few Body Syst. 54, 2427 (2013)]. This innovative approach, referred to as t-matrix-free in this paper, directly incorporates two-body (2B) interactions and completely avoids the 2B transition matrices. We extend this formalism to relativistic 3B bound states using a three-dimensional (3D) approach without using partial wave decomposition. To validate the proposed formulation, we perform a numerical study using spin-independent Malfliet–Tjon and Yamaguchi interactions. Our results demonstrate that the relativistic t-matrix-free Faddeev equation, which directly implements boosted interactions, accurately reproduces the 3B mass eigenvalues obtained from the conventional form of the Faddeev equation, referred to as t-matrix-dependent in this paper, with boosted 2B t-matrices. Moreover, the proposed formulation provides a simpler alternative to the standard approach, avoiding the computational complexity of calculating boosted 2B t-matrices and leading to significant computational time savings.more » « less
- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1007/s43036-024-00330-z
