Algebraic-delay differential systems: 𝐶⁰-extendable submanifolds and linearization
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Bibliographic record
Abstract
Consider the abstract algebraic-delay differential system, <disp-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row 1st Column x prime left-parenthesis t right-parenthesis 2nd Column a m p semicolon equals 3rd Column a m p semicolon upper A x left-parenthesis t right-parenthesis plus upper F left-parenthesis x left-parenthesis t right-parenthesis comma a left-parenthesis t right-parenthesis right-parenthesis comma 2nd Row 1st Column a left-parenthesis t right-parenthesis 2nd Column a m p semicolon equals 3rd Column a m p semicolon upper H left-parenthesis x Subscript t Baseline comma a Subscript t Baseline right-parenthesis period EndLayout"> <mml:semantics> <mml:mtable columnalign="right center left" rowspacing="3pt" columnspacing="0 thickmathspace" side="left" displaystyle="true"> <mml:mtr> <mml:mtd> <mml:msup> <mml:mi>x</mml:mi> <mml:mo>′</mml:mo> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <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:mtd> <mml:mtd> <mml:mi>a</mml:mi> <mml:mi>m</mml:mi> <mml:mi>p</mml:mi> <mml:mo>;</mml:mo> <mml:mi>A</mml:mi> <mml:mi>x</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>+</mml:mo> <mml:mi>F</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>,</mml:mo> <mml:mi>a</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">)</mml:mo> <mml:mo>,</mml:mo> </mml:mtd> </mml:mtr> <mml:mtr> <mml:mtd> <mml:mi>a</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <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:mtd> <mml:mtd> <mml:mi>a</mml:mi> <mml:mi>m</mml:mi> <mml:mi>p</mml:mi> <mml:mo>;</mml:mo> <mml:mi>H</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:mi>t</mml:mi> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>a</mml:mi> <mml:mi>t</mml:mi> </mml:msub> <mml:mo stretchy="false">)</mml:mo> <mml:mo>.</mml:mo> </mml:mtd> </mml:mtr> </mml:mtable> <mml:annotation encoding="application/x-tex">\begin{eqnarray*} x’(t) &=& Ax(t)+F(x(t),a(t)), \\ a(t) &=& H(x_t,a_t) . \end{eqnarray*}</mml:annotation> </mml:semantics> </mml:math> </disp-formula> Here <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A"> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding="application/x-tex">A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a linear operator on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper D left-parenthesis upper A right-parenthesis subset-of-or-equal-to upper X"> <mml:semantics> <mml:mrow> <mml:mi>D</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>A</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo> ⊆ </mml:mo> <mml:mi>X</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">D(A)\subseteq X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> satisfying the Hille-Yosida conditions, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x left-parenthesis t right-parenthesis element-of ModifyingAbove upper D left-parenthesis upper A right-parenthesis With bar subset-of-or-equal-to upper X"> <mml:semantics> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo> ∈ </mml:mo> <mml:mover> <mml:mrow> <mml:mi>D</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>A</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:mo accent="false"> ¯ </mml:mo> </mml:mover> <mml:mo> ⊆ </mml:mo> <mml:mi>X</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">x(t)\in \overline {D(A)}\subseteq X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="a left-parenthesis t right-parenthesis element-of bold upper R Superscript n"> <mml:semantics> <mml:mrow> <mml:mi>a</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo> ∈ </mml:mo> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">R</mml:mi> </mml:mrow> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">a(t)\in {\mathbf {R}}^n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , and
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 | 0.000 | 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 | 0.001 |
| 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