Periodic solutions of partial functional differential equations
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Bibliographic record
Abstract
In this paper we study the existence of periodic solutions to the partial functional differential equation <disp-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout Enlarged left-brace 1st Row StartFraction d y left-parenthesis t right-parenthesis Over d t EndFraction equals upper B y left-parenthesis t right-parenthesis plus ModifyingAbove upper L With caret left-parenthesis y Subscript t Baseline right-parenthesis plus f left-parenthesis t comma y Subscript t Baseline right-parenthesis comma for-all t greater-than-or-equal-to 0 comma 2nd Row y 0 equals phi element-of upper C Subscript upper B Baseline period EndLayout"> <mml:semantics> <mml:mrow> <mml:mo>{</mml:mo> <mml:mtable columnalign="left" rowspacing="4pt" columnspacing="1em"> <mml:mtr> <mml:mtd> <mml:mfrac> <mml:mrow> <mml:mi>d</mml:mi> <mml:mi>y</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:mrow> <mml:mi>d</mml:mi> <mml:mi>t</mml:mi> </mml:mrow> </mml:mfrac> <mml:mo>=</mml:mo> <mml:mi>B</mml:mi> <mml:mi>y</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>+</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mi>L</mml:mi> <mml:mo stretchy="false"> ^ </mml:mo> </mml:mover> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mi>y</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>t</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy="false">)</mml:mo> <mml:mo>+</mml:mo> <mml:mi>f</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>y</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>t</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy="false">)</mml:mo> <mml:mo>,</mml:mo> <mml:mspace width="thickmathspace"/> <mml:mi mathvariant="normal"> ∀ </mml:mi> <mml:mi>t</mml:mi> <mml:mo> ≥ </mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> </mml:mtd> </mml:mtr> <mml:mtr> <mml:mtd> <mml:msub> <mml:mi>y</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>0</mml:mn> </mml:mrow> </mml:msub> <mml:mo>=</mml:mo> <mml:mi> φ </mml:mi> <mml:mo> ∈ </mml:mo> <mml:msub> <mml:mi>C</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>B</mml:mi> </mml:mrow> </mml:msub> <mml:mo>.</mml:mo> </mml:mtd> </mml:mtr> </mml:mtable> <mml:mo fence="true" stretchy="true" symmetric="true"/> </mml:mrow> <mml:annotation encoding="application/x-tex">\begin{equation*} \left \{ \begin {array}{l} \frac {dy(t)}{dt}=By(t)+\hat {L}(y_{t})+f(t,y_{t}), \;\forall t\geq 0,\\ y_{0}=\varphi \in C_{B}. \end{array} \right . \end{equation*}</mml:annotation> </mml:semantics> </mml:math> </disp-formula> where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper B colon upper Y right-arrow upper Y"> <mml:semantics> <mml:mrow> <mml:mi>B</mml:mi> <mml:mo>:</mml:mo> <mml:mi>Y</mml:mi> <mml:mo stretchy="false"> → </mml:mo> <mml:mi>Y</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">B: Y \rightarrow Y</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a Hille-Yosida operator on a Banach space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Y"> <mml:semantics> <mml:mi>Y</mml:mi> <mml:annotation encoding="application/x-tex">Y</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . For <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C Subscript upper B Baseline colon-equal StartSet phi element-of upper C left-parenthesis left-bracket negative r comma 0 right-bracket semicolon upper Y right-parenthesis colon phi left-parenthesis 0 right-parenthesis element-of ModifyingAbove upper D left-parenthesis upper B right-parenthesis With bar EndSet"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>C</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>B</mml:mi> </mml:mrow> </mml:msub> <mml:mo>≔</mml:mo> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:mi> φ </mml:mi> <mml:mo> ∈ </mml:mo> <mml:mi>C</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mo stretchy="false">[</mml:mo> <mml:mo> − </mml:mo> <mml:mi>r</mml:mi> <mml:mo>,</mml:mo> <mml:mn>0</mml:mn> <mml:mo stretchy="false">]</mml:mo> <mml:mo>;</mml:mo> <mml:mi>Y</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>:</mml:mo> <mml:mi> φ </mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mn>0</mml:mn> <mml:mo stretchy="false">)</mml:mo>
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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 | 0.000 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.000 | 0.001 |
| 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) | 0.001 | 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