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Our preliminary investigation demonstrated a striking pattern in which balance variance decreased for balances closer to the tips of the phylogeny and increased for balances nearer to the root. To determine if this observed pattern was not the result of technical artifact, we took the following three steps. First, we omitted branch length weights from the transform as we anticipated that branch lengths may vary non-randomly as a function of depth in the phylogeny. Second, we anticipated that balances near the tips of the phylogeny would be associated with fewer read counts and thus would be more biased by our chosen heuristics for taxon weighting and zero replacement. We therefore omitted taxon weighting, employed more stringent filtering thresholds, and conditioned our calculation of balance variance on non-zero counts rather than using zero-replacement (Materials and methods). Third, we combined regression and a permutation scheme to test the null hypothesis that the degree to which neighboring clades covary is independent of the phylogenetic distance between them (Materials and methods). By permuting tip labels on the tree, our test generates a restricted subset of random sequential binary partitions that still maintains the count variability (and potential biases due to our zero handling methods) of the observed data.

With our modified PhILR analysis in place, we observed significantly decreasing balance variances near the tips of the phylogenetic tree for all body sites in the HMP dataset (p<0.01, permutation test with FDR correction; Figure 4A–F and Figure 4—figure supplements 1 –). Low variance balances predominated near the leaves of the tree. Examples of such balances involved B. fragilis species in stool ( Figure 4H ), Rothia mucilaginosa species in the buccal mucosa ( Figure 4J ), and Lactobacillus species in the mid-vagina ( Figure 4L ). By contrast, higher variance balances tended to be more basal on the tree. Three examples of high variance balances corresponded with clades at the order ( Figure 4G ), family ( Figure 4I ), and genus ( Figure 4K ) levels. We also observed that the relationship between balance variance and phylogenetic depth varied at different taxonomic scales. LOESS regression revealed that trends between variance and phylogenetic depth were stronger above the species level than below it (Materials and methods; Figure 4D–F and Figure 4—figure supplement 2 ). Overall, the observed pattern of decreasing balance variance near the tips of the phylogenetic tree suggested that closely related bacteria tend to covary in human body sites.

Figure 4 with 3 supplements

The variance of balance values captures the degree to which neighboring clades covary, with smaller balance variances representing sister clades that covary more strongly ( Figure 4—figure supplement …

Source code for Figure 4 and associated supplements.

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1 A sentence worded or expressed so as to elicit information.

enquiry , query
doubt , dispute, argument, debate, uncertainty, dubiousness, controversy, reservation
undoubted , beyond doubt, without doubt, certain, indubitable, indisputable, irrefutable, incontestable, incontrovertible, unquestionable, undeniable, unmistakable, clear, patent, manifest, obvious, palpable
indisputably , irrefutably, incontestably, incontrovertibly, unquestionably, undeniably, undoubtedly, beyond doubt, without doubt, certainly, indubitably, unmistakably, clearly, patently, manifestly, obviously, palpably

2 A matter requiring resolution or discussion.

subject , subject matter, theme, issue, matter, point, talking point, concern, argument, discussion, thesis, text, concept, field, area, keynote, leitmotif
issue , matter, business, problem, point at issue, point, concern, subject, topic, theme, item, case, proposal, proposition, debate, argument, dispute, bone of contention, controversy

1 Ask (someone) questions, especially in an official context.

interrogate , ask questions of, put questions to, cross-examine, cross-question, quiz, probe, canvass, catechize, interview, debrief, sound out, examine, give the third degree to
query , call in question, call into question, doubt, entertain doubts about, raise doubts about, throw doubt on, express suspicions about, harbour suspicions about, have suspicions about, suspect, feel uneasy about, express reservations about, harbour reservations about, have reservations about, challenge, dispute, cast aspersions on, object to, raise objections to

### Phrases

be a question of time

Be certain to happen sooner or later.

bring something into question

Raise an issue for further consideration or discussion.

come into question

Become an issue for further consideration or discussion.

good question

Said to indicate that one has been asked a particularly tricky question to which one does not know the answer.

in question

1 Being considered or discussed.

at issue , being discussed, under discussion, under consideration, on the agenda, for debate, to be discussed, to be decided

2 In doubt.

no question of

No possibility of.

out of the question

Too impracticable or unlikely to merit discussion.

impossible , beyond the bounds of possibility, impracticable, unattainable, unachievable, not feasible, not worth considering, unworkable, unobtainable, inconceivable, unthinkable, unimaginable, unrealizable, unsuitable

put the question

(in a formal debate or meeting) require supporters and opponents of a proposal to record their votes.

Origin

which in particular shows us that

Hence, we easily see that $5$ divides each coefficient of ${z}^{5m+1}$ in the infinite product expansions of

Finally, since we may write the generating function for the partition function as

we may equate the coefficients of ${z}^{5m+5}$ in the previous equations to prove our desired congruence result, namely that, $p\left(5m+4\right)\equiv 0\left(\mathrm{mod}5\right)$ for all $m\ge 0$ .

There are a number of transformations of generating functions that provide other applications (see the main article ). A transformation of a sequence's ordinary generating function (OGF) provides a method of converting the generating function for one sequence into a generating function enumerating another. These transformations typically involve integral formulas involving a sequence OGF (see integral transformations ) or weighted sums over the higher-order derivatives of these functions (see derivative transformations ).

Main article: Parental child abduction

By far the most common kind of child abduction is parental child abduction (200,000 in 2010 alone) [16] and often occurs when the parents separate or begin divorce proceedings. A parent may remove or retain the child from the other seeking to gain an advantage in expected or pending child-custody proceedings or because that parent fears losing the child in those expected or pending child-custody proceedings; a parent may refuse to return a child at the end of an access visit or may flee with the child to prevent an access visit or fear of domestic violence and abuse.

Parental child abductions may be within the same city, within the state region or within the same country, or may be international. Studies performed for the U.S. Department of Justice 's Office of Juvenile Justice and Delinquency Prevention reported that in 1999, 53% percent of family abducted children were gone less than one week, and 21% were gone one month or more. [17]

Main article: International child abduction

International child abduction occurs when a parent, relative or acquaintance of a child leaves the country with the child or children in violation of a custody decree or visitation order. Another related situation is retention where children are taken on an alleged vacation to a foreign country and are not returned.

While the number of cases which is over 600,000 a year consists of international child abduction is small in comparison to domestic cases, they are often the most difficult to resolve due to the involvement of conflicting international jurisdictions. Two-thirds of international parental abduction cases involve mothers who often allege domestic violence. Even when there is a treaty agreement for the return of a child, the court may be reluctant to return the child if the return could result in the permanent separation of the child from their primary caregiver. This could occur if the abducting parent faced criminal prosecution or deportation by returning to the child's home country.

The Hague Convention on the Civil Aspects of International Child Abduction is an international human rights treaty and legal mechanism to recover children abducted to another country. The Hague Convention does not provide relief in many cases resulting in some parents hiring private parties to recover their children. Covert recovery was first made public when Don Feeney, a former Delta Commando, responded to a desperate mother's plea to locate and recover her daughter from Jordan in the 1980s. Feeney successfully located and returned the child. A movie and book about Feeney's exploits lead to other desperate parents seeking him out for recovery services. [18]