Symmetric canonical quincunx tight framelets with high vanishing moments and smoothness
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Bibliographic record
Abstract
In this paper, we propose an approach to construct a family of two-dimensional compactly supported real-valued quincunx tight framelets <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-brace phi semicolon psi 1 comma psi 2 comma psi 3 right-brace"> <mml:semantics> <mml:mrow> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:mi> ϕ </mml:mi> <mml:mo>;</mml:mo> <mml:msub> <mml:mi> ψ </mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi> ψ </mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi> ψ </mml:mi> <mml:mn>3</mml:mn> </mml:msub> <mml:mo fence="false" stretchy="false">}</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\{\phi ; \psi _1,\psi _2,\psi _3\}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L 2 left-parenthesis double-struck upper R squared right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">L_2(\mathbb {R}^2)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with <italic>symmetry property</italic> and arbitrarily high orders of vanishing moments. Such quincunx tight framelets are associated with quincunx tight framelet filter banks <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-brace a semicolon b 1 comma b 2 comma b 3 right-brace"> <mml:semantics> <mml:mrow> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:mi>a</mml:mi> <mml:mo>;</mml:mo> <mml:msub> <mml:mi>b</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>b</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>b</mml:mi> <mml:mn>3</mml:mn> </mml:msub> <mml:mo fence="false" stretchy="false">}</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\{a;b_1,b_2,b_3\}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> having increasing orders of vanishing moments, possessing symmetry property, and enjoying the additional double canonical properties: <disp-formula content-type="math/mathml"> \[ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row 1st Column b 1 left-parenthesis k 1 comma k 2 right-parenthesis 2nd Column a m p semicolon equals left-parenthesis negative 1 right-parenthesis Superscript 1 plus k 1 plus k 2 Baseline a left-parenthesis 1 minus k 1 comma minus k 2 right-parenthesis comma 2nd Row 1st Column b 3 left-parenthesis k 1 comma k 2 right-parenthesis 2nd Column a m p semicolon equals left-parenthesis negative 1 right-parenthesis Superscript 1 plus k 1 plus k 2 Baseline b 2 left-parenthesis 1 minus k 1 comma minus k 2 right-parenthesis comma EndLayout for-all k 1 comma k 2 element-of double-struck upper Z period"> <mml:semantics> <mml:mrow> <mml:mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" side="left" displaystyle="true"> <mml:mtr> <mml:mtd> <mml:msub> <mml:mi>b</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mi>k</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>k</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo stretchy="false">)</mml:mo> </mml:mtd> <mml:mtd> <mml:mi>a</mml:mi> <mml:mi>m</mml:mi> <mml:mi>p</mml:mi> <mml:mo>;</mml:mo> <mml:mo>=</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mo> − </mml:mo> <mml:mn>1</mml:mn> <mml:msup> <mml:mo stretchy="false">)</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>1</mml:mn> <mml:mo>+</mml:mo> <mml:msub> <mml:mi>k</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>+</mml:mo> <mml:msub> <mml:mi>k</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:mrow> </mml:msup> <mml:mi>a</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mn>1</mml:mn> <mml:mo> − </mml:mo> <mml:msub> <mml:mi>k</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:mo> − </mml:mo> <mml:msub> <mml:mi>k</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo stretchy="false">)</mml:mo> <mml:mo>,</mml:mo> </mml:mtd> </mml:mtr> <mml:mtr> <mml:mtd> <mml:msub> <mml:mi>b</mml:mi> <mml:mn>3</mml:mn> </mml:msub> <mml:mo stretch
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
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 | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 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