Why this work is in the frame
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Bibliographic record
Abstract
Abstract The distribution of monomers along a linear polymer grafted on a hard wall is modelled by determining the probability distribution of occupied vertices of Dyck path and Dyck meander models of adsorbing linear polymers. For example, the probability that a Dyck path passes through the lattice site with coordinates <?CDATA $(\lfloor \epsilon n\rfloor ,\lfloor \delta \sqrt{n}\rfloor )$?> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mo stretchy="false">(</mml:mo> <mml:mo stretchy="true">⌊</mml:mo> <mml:mi>ϵ</mml:mi> <mml:mi>n</mml:mi> <mml:mo stretchy="true">⌋</mml:mo> <mml:mo>,</mml:mo> <mml:mo stretchy="true">⌊</mml:mo> <mml:mi>δ</mml:mi> <mml:msqrt> <mml:mrow> <mml:mi>n</mml:mi> </mml:mrow> </mml:msqrt> <mml:mo stretchy="true">⌋</mml:mo> <mml:mo stretchy="false">)</mml:mo> </mml:math> in the square lattice, for 0 < ϵ < 1 and δ ≥ 0, is determined asymptotically as n → ∞ and this uncovers the probability density of vertices along Dyck paths in the limit as the length of the path n approaches infinity: <?CDATA \begin{eqnarray*}{{\mathbb{P}}}^{(D)}(\epsilon ,\delta )=\displaystyle \frac{4{\delta }^{2}}{\sqrt{\pi \,{\epsilon }^{3}{\left(1-\epsilon \right)}^{3}}}\,{e}^{-{\delta }^{2}/\epsilon (1-\epsilon )}.\end{eqnarray*}?> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msup> <mml:mrow> <mml:mi mathvariant="double-struck">P</mml:mi> </mml:mrow> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>D</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mi>ϵ</mml:mi> <mml:mo>,</mml:mo> <mml:mi>δ</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mstyle displaystyle="true"> <mml:mfrac> <mml:mrow> <mml:mn>4</mml:mn> <mml:msup> <mml:mrow> <mml:mi>δ</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> <mml:mrow> <mml:msqrt> <mml:mrow> <mml:mi>π</mml:mi> <mml:mspace width="0.25em" /> <mml:msup> <mml:mrow> <mml:mi>ϵ</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>3</mml:mn> </mml:mrow> </mml:msup> <mml:msup> <mml:mrow> <mml:mfenced close=")" open="("> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>−</mml:mo> <mml:mi>ϵ</mml:mi> </mml:mrow> </mml:mfenced> </mml:mrow> <mml:mrow> <mml:mn>3</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> </mml:msqrt> </mml:mrow> </mml:mfrac> </mml:mstyle> <mml:mspace width="0.25em" /> <mml:msup> <mml:mrow> <mml:mi>e</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>−</mml:mo> <mml:msup> <mml:mrow> <mml:mi>δ</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> <mml:mrow> <mml:mo stretchy="true">/</mml:mo> </mml:mrow> <mml:mi>ϵ</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>−</mml:mo> <mml:mi>ϵ</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:msup> <mml:mi>.</mml:mi> </mml:math> The properties of a polymer coating of a hard wall and the density or distribution of monomers in the coating is relevant in applications such as the stabilisation of a colloid dispersion by a polymer or in a drug delivery system such as a drug-eluding stent covered by a grafted polymer.
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) | 0.000 | 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