A randomly perturbed graph G^p = G_α∪ G_n,p is obtained by taking a deterministic n-vertex graph G_α= (V, E) with minimum degree δ(G)≥ αn and adding the edges of the binomial random graph G_n,p defined on the same vertex set V. For which value p (depending on α) does the graph G^p contain a K_r-factor – a spanning collection of vertex-disjoint copies of K_r – with high probability? The order of magnitude of the minimum such p was determined whenever α≠ 1- s/r for an integer s by Balogh, Treglown and Wagner, and by Han, Morris and Treglown. In earlier work, the first three authors determined this threshold probability p_s up to a constant factor for all values of α= 1-s/r≤1/2. Here, we complete the picture by establishing p_s in the remaining case α> 1/2. A key ingredient in our approach is an extremal result of independent interest: we prove a fractional stability version of a tiling theorem due to Shokoufandeh and Zhao.
An explicit formula for the prime-counting function π(x), usually attributed to Riemann and von Mangoldt, is prominently stated as the equation π(x)=R(x)-∑_ρR(x^ρ), where the sum runs over all zeros ρ of the Riemann ζ-function, the non-trivial ones being ordered by increasing absolute value of their imaginary parts and counted with multiplicity. This particularly entails the claim that the partial sums over the non-trivial zeros, ΣR_T(x):=∑_0<| m(ρ)|≤ T R(x^ρ) converge as T→∞. Writing Θ:=sup{ e(ρ): ζ(ρ)=0, 0< e(ρ)<1}, for what has recently been called “Riemann's constant”, we prove that, for every fixed x>1 and every θ<Θ, the sums ΣR_T(x) are not O(T^θ). As a consequence, lim sup_T→∞|ΣR_T(x)|=∞ and ∑_ρR(x^ρ) diverges. We conclude the paper by showing that an adapted, but simpler strategy also gives the divergence of the contribution of the trivial zeros to ∑_ρR(x^ρ).
Background Biomedicine has built competent tools for appraising results. Yet a finished, technically sound result is routinely reported and then used to support a broader claim, one that would need a comparison the study never made. Methods This is a conceptual analysis and coverage map of the major appraisal tools and validity concepts, including the risk-of-bias tools, causal diagrams, estimands, target-trial emulation, GRADE indirectness, transportability, clinical-applicability frameworks, outcome-reporting extensions, trial-trustworthiness checks and the classical validity tradition. Each is placed on four axes: whether it is used forwards or backwards, the level at which it works, its reference point, and its design scope. Five published trials were chosen purposively to show distinct ways a result and its interpretation come apart. Results The mapped tools cover estimate fidelity, causal identification, question specification, outcome anchoring, evidence transfer, clinical applicability, trustworthiness screening and the classical validity concerns. Their borders do not converge on one point: the finished result read against the specific interpretation now taken from it. There a result can be unbiased, identifiable, well specified, applicable to a fixed target, clinically appraisable and trustworthy as a report, and still be cited for a comparison it never made. PLCO, NordICC, Look AHEAD, CIRT and SPRINT trials come apart in five different ways: a contaminated comparator, an invitation most declined, a programme mistaken for its target, a pathway never engaged, and a measurement regime left behind. In every case the feature that fixes the contrast the study actually made was present in the published report, and unused. Conclusions Existing tools can appraise results in great detail. However, none of them routinely asks whether the interpretation now attached to the result is one its actual comparison can support. Construct, external and scope validity named this territory, but naming is not an operation. The check is simple: read what the study actually compared, then ask whether the interpretation now placed on it needs more than that. The gap is not in the evidence but in the reading of it.
Biomimetic design is often justified by the claim that evolution has refined biological systems under severe selective pressure; however, this claim is incomplete. Evolution does not produce optimal solutions, but constrained trade-off resolutions. The translational question is therefore not whether a biological system performs the desired function, but whether the functional principle can survive separation from the system that produced it. Convergent evolution, where distantly related lineages independently arrive at similar solutions to the same functional problem, raises the probability that such solutions reflect physical or chemical constraints, which are stronger candidates for transfer into biomaterial design. Lineage-isolated solutions require a different test, namely whether the function reduces to a feature that can be reproduced outside the source organism. The argument is demonstrated through a convergence × reducibility matrix and an ex natura protocol from a biological phenomenon to a testable biomaterial claim. Biomimetics earns its place not as a universal design doctrine, but in those situations where evolutionary trade-off resolutions can survive translation into safe and manufacturable biomaterials.
Biomimetics often treats convergent evolution as the strongest sign that a biological solution is general. That inference is safest when the constraint does not counter-adapt. Hosts do. In coevolved host-interface systems, recurrence alone cannot tell us whether a solution is translatable. Biomimetic transferability depends first on two axes: conservation of the host target and versatility of the attacking lineage. A conserved target behaves, for translational purposes, like a biochemical constraint, a broad-host parasite has already tested its mechanism across biological variation. The window narrows when the target is taxonomically local, or when the mechanism has become a private molecular conversation inside a narrow dyad. Haematophagous feeders, intracellular protozoans, specialist helminths, and polydnavirus-bearing parasitoid wasps therefore do not offer the same kind of biomimetic object. Some yield molecules, some vulnerability maps, some contextual principles, and some only architecture or analogy. The point is not to mine coevolved systems less, but to stop mistaking coevolutionary success for biomimetic portability.
