Matrix Extension with Symmetry and Its Application to Symmetric Orthonormal Multiwavelets
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Bibliographic record
Abstract
Let $\mathsf{P}$ be an $r\times s$ matrix of Laurent polynomials with symmetry such that $\mathsf{P}(z)\mathsf{P}^*(z)=I_r$ for all $z\in\mathbb{C}\backslash\{0\}$ and the symmetry of $\mathsf{P}$ is compatible. The matrix extension problem with symmetry is to find an $s\times s$ square matrix $\mathsf{P}_e$ of Laurent polynomials with symmetry such that $[I_r,\mathbf{0}]\mathsf{P}_e =\mathsf{P}$ (that is, the submatrix of the first r rows of $\mathsf{P}_e$ is the given matrix $\mathsf{P}$), $\mathsf{P}_e$ is paraunitary satisfying $\mathsf{P}_e(z)\mathsf{P}_e^*(z)=I_s$ for all $z\in\mathbb{C}\backslash\{0\}$, and the symmetry of $\mathsf{P}_e$ is compatible. Moreover, it is highly desirable in many applications that the support of the coefficient sequence of $\mathsf{P}_e$ can be controlled by that of $\mathsf{P}$. In this paper, we completely solve the matrix extension problem with symmetry by constructing such a desired matrix $\mathsf{P}_e$ from a given matrix $\mathsf{P}$. Furthermore, using a cascade structure, we obtain a complete representation of any $r\times s$ paraunitary matrix $\mathsf{P}$ having compatible symmetry, which in turn leads to a construction of a desired matrix $\mathsf{P}_e$ from a given matrix $\mathsf{P}$. Matrix extension plays an important role in many areas such as wavelet analysis, electronic engineering, system sciences, and so on. As an application of our general results on matrix extension with symmetry, we obtain a satisfactory algorithm for constructing symmetric orthonormal multiwavelets by deriving high-pass filters with symmetry from any given orthogonal low-pass filters with symmetry. Several examples of symmetric orthonormal multiwavelets are provided to illustrate the results in this paper.
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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.003 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.004 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.001 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it