Unlike humans, some birds have independently evolved to flourish on sugar-rich nectar &fruit without ill effect. In a new Science study, researchers find that these bird species share convergent evolutionary changes in key physiological traits and metabolic genes that enable their high-sugar diets.

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The effectively tells us that the middle term — ⌈r · b⌉ - 1 — is always less than the value needed to match r, but the difference is never greater than a single tick of the numerator. We’re as close as we can be; the value of a for the optimal low-side approximation (a / b

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