Singular value
View article: Perron-Frobenius Theory Beyond Positivity: Spectral Geometry via Singular Values
Perron-Frobenius Theory Beyond Positivity: Spectral Geometry via Singular Values Open
The Perron-Frobenius theorem is a cornerstone of linear algebra, providing profound insights into the spectral properties of non-negative matrices. Its utility spans various fields, from economics and demographics to graph theory and Marko…
View article: Perron-Frobenius Theory Beyond Positivity: Spectral Geometry via Singular Values
Perron-Frobenius Theory Beyond Positivity: Spectral Geometry via Singular Values Open
The Perron-Frobenius theorem is a cornerstone of linear algebra, providing profound insights into the spectral properties of non-negative matrices. Its utility spans various fields, from economics and demographics to graph theory and Marko…
View article: Beyond Canonical Forms: Spectral Rigidity and Geometric Invariants of Matrix Decompositions
Beyond Canonical Forms: Spectral Rigidity and Geometric Invariants of Matrix Decompositions Open
This paper delves into the spectral properties of matrix decompositions, specifically focusing on spectral rigidity and geometric invariants beyond the limitations of canonical forms. We investigate how spectral information can be leverage…
View article: The Spectral Geometry of Tensor Rank
The Spectral Geometry of Tensor Rank Open
The concept of tensor rank, a fundamental measure of complexity for multi-dimensional arrays, presents significant challenges in both its computation and theoretical understanding. While matrix rank is well-understood through singular valu…
View article: Fractal Regularity and the Fine Structure of Singular Measures
Fractal Regularity and the Fine Structure of Singular Measures Open
This paper explores the intricate relationship between fractal regularity and the fine structure of singular measures. We investigate how the fractal dimension of a measure's support and its local scaling properties influence its regularit…
View article: The Spectral Geometry of Tensor Rank
The Spectral Geometry of Tensor Rank Open
The concept of tensor rank, a fundamental measure of complexity for multi-dimensional arrays, presents significant challenges in both its computation and theoretical understanding. While matrix rank is well-understood through singular valu…
View article: CR Foliations and the Levi Problem in Singular Complex Spaces
CR Foliations and the Levi Problem in Singular Complex Spaces Open
This paper investigates the interplay between CR foliations and the Levi problem in the context of singular complex spaces. We extend classical results from complex manifold theory to singular settings, focusing on the existence and proper…
View article: CR Foliations and the Levi Problem in Singular Complex Spaces
CR Foliations and the Levi Problem in Singular Complex Spaces Open
This paper investigates the interplay between CR foliations and the Levi problem in the context of singular complex spaces. We extend classical results from complex manifold theory to singular settings, focusing on the existence and proper…
View article: Beyond Canonical Forms: Spectral Rigidity and Geometric Invariants of Matrix Decompositions
Beyond Canonical Forms: Spectral Rigidity and Geometric Invariants of Matrix Decompositions Open
This paper delves into the spectral properties of matrix decompositions, specifically focusing on spectral rigidity and geometric invariants beyond the limitations of canonical forms. We investigate how spectral information can be leverage…
View article: Fast orbit feedback using the GSVD for systems with multiple slow corrector arrays
Fast orbit feedback using the GSVD for systems with multiple slow corrector arrays Open
Advances in detector speed and resolution at 4th generation light sources make electron beam stability a critical requirement. At Diamond-II, the fast orbit feedback (FOFB) will stabilise the beam using 252 beam position monitors and two a…
View article: Fractal Regularity and the Fine Structure of Singular Measures
Fractal Regularity and the Fine Structure of Singular Measures Open
This paper explores the intricate relationship between fractal regularity and the fine structure of singular measures. We investigate how the fractal dimension of a measure's support and its local scaling properties influence its regularit…
View article: Solving sparsity and scalability problems for book recommendations on e-commerce
Solving sparsity and scalability problems for book recommendations on e-commerce Open
This study proposed a hierarchical density-based spatial clustering of applications with noise (HDBSCAN) and randomized singular value decomposition (RSVD) collaborative filtering (CF) method to overcome sparsity and scalability problems f…
View article: The Dual Nature of Prime Gaps
The Dual Nature of Prime Gaps Open
1️⃣ Description – PDF File: The Dual Nature of Prime Gaps.pdf This preprint investigates the statistical structure of prime gaps through a dual lens: Random Matrix Theory on the one hand, and the Hardy–Littlewood singular series (Prime Cos…
View article: The Dual Nature of Prime Gaps
The Dual Nature of Prime Gaps Open
1️⃣ Description – PDF File: The Dual Nature of Prime Gaps.pdf This preprint investigates the statistical structure of prime gaps through a dual lens: Random Matrix Theory on the one hand, and the Hardy–Littlewood singular series (Prime Cos…
View article: Necessary and Sufficient Criterion for Singular or Nonsingular of Diagonally Dominant Matrices
Necessary and Sufficient Criterion for Singular or Nonsingular of Diagonally Dominant Matrices Open
The problem of determining whether a diagonally dominant matrix is singular or nonsingular is a classical topic in matrix theory. This paper develops necessary and sufficient conditions for the singularity or nonsingularity of diagonally d…
View article: Weyl distributions, spectral properties, and circulant approximation results for quaternion block multilevel Toeplitz matrix sequences
Weyl distributions, spectral properties, and circulant approximation results for quaternion block multilevel Toeplitz matrix sequences Open
The present work contains a comprehensive treatment of Weyl eigenvalue and singular value distributions for single-axis quaternion block multilevel Toeplitz matrix sequences generated by $s\times t$ quaternion matrix-valued, $d$-variate, L…
View article: Weyl distributions, spectral properties, and circulant approximation results for quaternion block multilevel Toeplitz matrix sequences
Weyl distributions, spectral properties, and circulant approximation results for quaternion block multilevel Toeplitz matrix sequences Open
The present work contains a comprehensive treatment of Weyl eigenvalue and singular value distributions for single-axis quaternion block multilevel Toeplitz matrix sequences generated by $s\times t$ quaternion matrix-valued, $d$-variate, L…
View article: An SSA-SARIMA-GSVR Hybrid Model Based on Singular Spectrum Analysis for O3-CPM Prediction
An SSA-SARIMA-GSVR Hybrid Model Based on Singular Spectrum Analysis for O3-CPM Prediction Open
Ozone density at cold-point mesopause (O3-CPM) can provide information on long-term atmospheric trends. Compared to ground-level ozone, O3-CPM is not only adversely affected by chemical substances emitted from human activities but is also …
View article: Singular extremals of optimal control problems with $L^1$ cost
Singular extremals of optimal control problems with $L^1$ cost Open
We study the optimal control problem for a control-affine system, where we want to minimize the $L^1$ norm of the control. First, we show how Pontryagin Maximum Principle (PMP) applies to this problem and we divide the extremal trajectorie…
View article: Effective Hyper-clutter Artifacts Suppression for Ultrafast Ultrasound Doppler Imaging
Effective Hyper-clutter Artifacts Suppression for Ultrafast Ultrasound Doppler Imaging Open
Objective: Hyper-clutter artifacts (HCA), arising from strong tissue reflections or physiological motion, present persistent challenges in ultrafast ultrasound Doppler imaging, often obscuring surrounding small vessel flow signals, especia…
View article: Effective Hyper-clutter Artifacts Suppression for Ultrafast Ultrasound Doppler Imaging
Effective Hyper-clutter Artifacts Suppression for Ultrafast Ultrasound Doppler Imaging Open
Objective: Hyper-clutter artifacts (HCA), arising from strong tissue reflections or physiological motion, present persistent challenges in ultrafast ultrasound Doppler imaging, often obscuring surrounding small vessel flow signals, especia…
View article: Universal Renormalization of Coupled Singular SPDE Systems
Universal Renormalization of Coupled Singular SPDE Systems Open
Stochastic Partial Differential Equations (SPDEs) are indispensable tools for modeling complex systems subjected to random influences, ranging from statistical physics and fluid dynamics to finance and biology. A significant challenge aris…
View article: Perron-Singular Decompositions of Irreducible Rectangular Matrices
Perron-Singular Decompositions of Irreducible Rectangular Matrices Open
The Perron-Frobenius theorem is a cornerstone of non-negative matrix theory, describing the existence and uniqueness of a dominant positive eigenvalue and corresponding eigenvector for irreducible non-negative square matrices. This theorem…
View article: Universal Renormalization of Coupled Singular SPDE Systems
Universal Renormalization of Coupled Singular SPDE Systems Open
Stochastic Partial Differential Equations (SPDEs) are indispensable tools for modeling complex systems subjected to random influences, ranging from statistical physics and fluid dynamics to finance and biology. A significant challenge aris…
View article: Perron-Singular Decompositions of Irreducible Rectangular Matrices
Perron-Singular Decompositions of Irreducible Rectangular Matrices Open
The Perron-Frobenius theorem is a cornerstone of non-negative matrix theory, describing the existence and uniqueness of a dominant positive eigenvalue and corresponding eigenvector for irreducible non-negative square matrices. This theorem…
View article: Matricial Gaussian quadrature rules: singular case
Matricial Gaussian quadrature rules: singular case Open
Let $L$ be a linear operator on univariate polynomials of bounded degree taking values in real symmetric matrices, whose moment matrix is positive semidefinite. Assume that $L$ admits a positive matrix-valued representing measure $μ$. Any …
View article: Universal scaling limits at the spectral singularity of structured random matrices
Universal scaling limits at the spectral singularity of structured random matrices Open
The empirical spectral distribution of Hermitian $K \times K$-block random matrices converges to a deterministic density on the real line with a potential atom at the origin as the dimension of the blocks tends to infinity. In this model t…
View article: Universal scaling limits at the spectral singularity of structured random matrices
Universal scaling limits at the spectral singularity of structured random matrices Open
The empirical spectral distribution of Hermitian $K \times K$-block random matrices converges to a deterministic density on the real line with a potential atom at the origin as the dimension of the blocks tends to infinity. In this model t…
View article: Matricial Gaussian quadrature rules: singular case
Matricial Gaussian quadrature rules: singular case Open
Let $L$ be a linear operator on univariate polynomials of bounded degree taking values in real symmetric matrices, whose moment matrix is positive semidefinite. Assume that $L$ admits a positive matrix-valued representing measure $μ$. Any …
View article: Low‐Rank SPIKE Framework for Solving Large Sparse Linear Systems With Applications
Low‐Rank SPIKE Framework for Solving Large Sparse Linear Systems With Applications Open
The SPIKE family of linear system solvers provides parallelism using a block tridiagonal partitioning. Typically SPIKE‐based solvers are applied to banded systems, resulting in structured off‐diagonal blocks with nonzeros elements restrict…