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# Ancient DNA from Chalcolithic Israel reveals the role of population mixture in cultural transformation

## Abstract

The material culture of the Late Chalcolithic period in the southern Levant (4500–3900/3800 BCE) is qualitatively distinct from previous and subsequent periods. Here, to test the hypothesis that the advent and decline of this culture was influenced by movements of people, we generated genome-wide ancient DNA from 22 individuals from Peqi’in Cave, Israel. These individuals were part of a homogeneous population that can be modeled as deriving ~57% of its ancestry from groups related to those of the local Levant Neolithic, ~17% from groups related to those of the Iran Chalcolithic, and ~26% from groups related to those of the Anatolian Neolithic. The Peqi’in population also appears to have contributed differently to later Bronze Age groups, one of which we show cannot plausibly have descended from the same population as that of Peqi’in Cave. These results provide an example of how population movements propelled cultural changes in the deep past.

## Introduction

The material culture of the Late Chalcolithic period in the southern Levant contrasts qualitatively with that of earlier and later periods in the same region. The Late Chalcolithic in the Levant is characterized by increases in the density of settlements, introduction of sanctuaries1,2,3, utilization of ossuaries in secondary burials4,5, and expansion of public ritual practices as well as an efflorescence of symbolic motifs sculpted and painted on artifacts made of pottery, basalt, copper, and ivory6,7,8,9. The period’s impressive metal artifacts, which reflect the first known use of the “lost wax” technique for casting of copper, attest to the extraordinary technical skill of the people of this period10,11.

The distinctive cultural characteristics of the Late Chalcolithic period in the Levant (often related to the Ghassulian culture, although this term is not in practice applied to the Galilee region where this study is based) have few stylistic links to the earlier or later material cultures of the region, which has led to extensive debate about the origins of the people who made this material culture. One hypothesis is that the Chalcolithic culture in the region was spread in part by immigrants from the north (i.e., northern Mesopotamia), based on similarities in artistic designs12,13. Others have suggested that the local populations of the Levant were entirely responsible for developing this culture, and that any similarities to material cultures to the north are due to borrowing of ideas and not to movements of people2,14,15,16,17,18,19.

To explore these questions, we studied ancient DNA from a Chalcolithic site in Northern Israel, Peqi’in (Fig. 1a). This cave, which is around 17 m long and 4.5–8.0 m wide (Fig. 1b), was discovered during road construction in 1995, and was sealed by natural processes during or around the end of the Late Chalcolithic period (around 3900 BCE). Archeological excavations have revealed an extraordinary array of finely crafted objects, including chalices, bowls, and churns, as well as more than 200 ossuaries and domestic jars repurposed as ossuaries (the largest number ever found in a single cave), often decorated with anthropomorphic designs (Fig. 1c)20,21. It has been estimated that the burial cave contained up to 600 individuals22, making it the largest burial site ever identified from the Late Chalcolithic period in the Levant. Direct radiocarbon dating suggests that the cave was in use throughout the Late Chalcolithic (4500–3900 BCE), functioning as a central burial location for the region21,23.

Previous genome-wide ancient DNA studies from the Near East have revealed that at the time when agriculture developed, populations from Anatolia, Iran, and the Levant were approximately as genetically differentiated from each other as present-day Europeans and East Asians are today24,25. By the Bronze Age, however, expansion of different Near Eastern agriculturalist populations—Anatolian, Iranian, and Levantine—in all directions and admixture with each other substantially homogenized populations across the region, thereby contributing to the relatively low genetic differentiation that prevails today24. Lazaridis et al.24 showed that the Levant Bronze Age population from the site of 'Ain Ghazal, Jordan (2490–2300 BCE) could be fit statistically as a mixture of around 56% ancestry from a group related to Levantine Pre-Pottery Neolithic agriculturalists (represented by ancient DNA from Motza, Israel and 'Ain Ghazal, Jordan; 8300–6700 BCE) and 44% related to populations of the Iranian Chalcolithic (Seh Gabi, Iran; 4680–3662 calBCE). Haber et al.26 suggested that the Canaanite Levant Bronze Age population from the site of Sidon, Lebanon (~1700 BCE) could be modeled as a mixture of the same two groups albeit in different proportions (48% Levant Neolithic-related and 52% Iran Chalcolithic-related). However, the Neolithic and Bronze Age sites analyzed so far in the Levant are separated in time by more than three thousand years, making the study of samples that fill in this gap, such as those from Peqi’in, of critical importance.

In a dedicated clean room facility at Harvard Medical School, we obtained bone powder from 48 skeletal remains, of which 37 were petrous bones known for excellent DNA preservation27. We extracted DNA28 and built next-generation sequencing libraries to which we attached unique barcodes to minimize the possibility of contamination. We treated the libraries with Uracil–DNA glycosylase (UDG) to reduce characteristic ancient DNA damage at all but the first and last nucleotides29 (Supplementary Table 1 and Supplementary Data 1 provide background for successful samples and report information for each library, respectively). After initial screening by enriching the libraries for mitochondrial DNA, we enriched promising libraries for sequences overlapping about 1.2 million single nucleotide polymorphisms (SNPs)30,31. We evaluated each individual for evidence of authentic ancient DNA by limiting to libraries with a minimum of 3% cytosine-to-thymine errors at the final nucleotide29, by requiring that the ratio of X-to-Y-chromosome sequences was characteristic of either a male or a female, by requiring >95% matching to the consensus sequence of mitochondrial DNA30, and by requiring (for males) a lack of variation at known polymorphic positions on chromosome X (point estimates of contamination of less than 2%)32. We also restricted to individuals with at least 5000 of the targeted SNPs covered at least once.

This procedure produced genome-wide data from 22 ancient individuals from Peqi’in Cave (4500–3900 calBCE), with the individuals having a median of 358,313 of the targeted SNPs covered at least once (range: 25,171–1,002,682). The dataset is of exceptional quality given the typically poor preservation of DNA in the warm Near East, with a higher proportion of samples yielding appreciable coverage of ancient DNA than has previously been obtained from the region, likely reflecting the optimal sampling techniques we used and the favorable preservation conditions at the cave. We analyzed this dataset in conjunction with previously published datasets of ancient Near Eastern populations24,26 to shed light on the history of the individuals buried in the Peqi’in cave site, and on the population dynamics of the Levant during the Late Chalcolithic period.

## Results

### Genetic differentiation and diversity in the ancient Levant

A total of 20 Peqi’in samples appear to be unrelated to each other to the limits of our resolution (that is, genetic analysis suggested that they were not first, second, or third degree relatives of each other), and we used these as our analysis set. Taking advantage of the new data point added by the Peqi’in samples, we began by studying how genetic differentiation among Levantine populations changed over time. We replicate previous reports of dramatic decline in genetic differentiation over time in West Eurasia24, observing a median pairwise FST of 0.023 (range: 0.009–0.061) between the Peqi’in samples (abbreviation: Levant_ChL) and other West Eurasian Neolithic and Chalcolithic populations, relative to a previously reported median pairwise FST of 0.098 (range: 0.023–0.153) observed between populations in pre-Neolithic periods, 0.015 (range: 0.002–0.045) in the Bronze Age periods, and 0.011 (range: 0–0.046) in present-day West Eurasian populations24. Thus, the collapse to present-day levels of differentiation was largely complete by the Chalcolithic (Supplementary Figure 1).

We also observe an increase in genetic diversity over time in the Levant as measured by the rate of polymorphism between two random genome sequences at each SNP analyzed in our study. Specifically, the Levant_ChL population exhibits an intermediate level of heterozygosity relative to the earlier and later populations (Fig. 2).

Both the increasing genetic diversity over time, and the reduced differentiation between populations as measured via FST, are consistent with a model in which gene flow reduced differentiation across groups while increasing diversity within groups.

### Genetic affinities of the individuals of Peqi’in Cave

To obtain a qualitative picture of how these individuals relate to previously published ancient DNA and to present-day people, we began by carrying out principal component analysis (PCA)33. In a plot of the first and second principal components (Fig. 3a), the samples from Peqi’in Cave form a tight cluster, supporting the grouping of these individuals into a single analysis population (while we use the broad name “Levant_ChL” to refer to these samples, we recognize that they are currently the only ancient DNA available from the Levant in this time period and future work will plausibly reveal genetic substructure in Chalcolithic samples over the broad region). The Levant_ChL cluster overlaps in the PCA with a cluster containing Neolithic Levantine samples (Levant_N), although it is slightly shifted upward on the plot toward a cluster corresponding to samples from the Levant Bronze Age, including samples from 'Ain Ghazal, Jordan (Levant_BA_South) and Sidon, Lebanon (Levant_BA_North). The placement of the Levant_ChL cluster is consistent with a previously observed pattern whereby chronologically later Levantine populations are shifted towards the Iran Chalcolithic (Iran_ChL) population compared to earlier Levantine populations, Levant_N (Pre-Pottery and Pottery Neolithic agriculturalists from present-day Israel and Jordan) and Natufians (Epipaleolithic hunter-gatherers from present-day Israel)24.

ADMIXTURE model-based clustering analyses34 produced results consistent with PCA in suggesting that individuals from the Levant_ChL population had a greater affinity on average to Iranian agriculturalist-related populations than was the case for earlier Levantine individuals. Figure 3b shows the ADMIXTURE results for the ancient individuals assuming K = 11 clusters (we selected this number because it maximizes ancestry components that are correlated to ancient populations from the Levant, from Iran, and European hunter-gatherers)24. Like all Levantine populations, the primary ancestry component assigned to the Levant_ChL population, shown in blue, is maximized in earlier Levant_N and Natufian individuals. ADMIXTURE also assigns a component of ancestry in Levant_ChL, shown in green, to a population that is generally absent in the earlier Levant_N and Natufian populations, but is present in later Levant_BA_South and Levant_BA_North samples. This green component is also inferred in small proportions in several samples assigned to the Levant_N, but there is not a clear association to archaeological location or date, and these individuals are not significantly genetically distinct from the other individuals included in Levant_N by formal testing, and thus we pool all Levant_N for the primary analyses in this study (Supplementary Note 1)24.

### Population continuity and admixture in the Levant

To determine the relationship of the Levant_ChL population to other ancient Near Eastern populations, we used f-statistics35 (see Supplementary Note 2 for more details). We first evaluated whether the Levant_ChL population is consistent with descending directly from a population related to the earlier Levant_N. If this was the case, we would expect that the Levant_N population would be consistent with being more closely related to the Levant_ChL population than it is to any other population, and indeed we confirm this by observing positive statistics of the form f4 (Levant_ChL, A; Levant_N, Chimpanzee) for all ancient test populations, A (Fig. 4a). However, Levant_ChL and Levant_N population do not form a clade, as when we compute symmetry statistics of the form f4 (Levant_N, Levant_ChL; A, Chimpanzee), we find that the statistic is often negative, with Near Eastern populations outside the Levant sharing more alleles with Levant_ChL than with Levant_N (Fig. 4b). We conclude that while the Levant_N and Levant_ChL populations are clearly related, the Levant_ChL population cannot be modeled as descending directly from the Levant_N population without additional admixture related to ancient Iranian agriculturalists. Direct evidence that Levant_ChL is admixed comes from the statistic f3 (Levant_ChL; Levant_N, A), which for some populations, A, is significantly negative indicating that allele frequencies in Levant_ChL tend to be intermediate between those in Levant_N and A—a pattern that can only arise if Levant_ChL is the product of admixture between groups related, perhaps distantly, to Levant_N and A35. The most negative f3- and f4-statistics are produced when A is a population from Iran or the Caucasus. This suggests that the Levant_ChL population is descended from a population related to Levant_N, but also harbors ancestry from non-Levantine populations related to those of Iran or the Caucasus that Levant_N does not share (or at least share to the same extent).

### The ancestry of the Levant Chalcolithic people

We used qpAdm as our main tool for identifying plausible admixture models for the ancient populations for which we have data (see Supplementary Note 3 for more details)36.

The qpAdm method evaluates whether a tested set of N “Left” populations—including a “target” population (the population whose ancestry is being modeled) and a set of N − 1 additional populations—are consistent with being derived from mixtures in various proportions of N − 1 ancestral populations related differentially to a set of outgroup populations, referred to as “Right” populations. For all our analyses, we use a base set of 11 “Right” outgroups referred to collectively as “09NW”—Ust_Ishim, Kostenki14, MA1, Han, Papuan, Onge, Chukchi, Karitiana, Mbuti, Natufian, and WHG—whose value for disentangling divergent strains of ancestry present in ancient Near Easterners has been documented in Lazaridis et al.24 (for some analyses we supplement this set with additional outgroups). To evaluate whether the “Left” populations are consistent with a hypothesis of being derived from N − 1 sources, qpAdm effectively computes all possible statistics of the form f4(Lefti, Leftj; Rightk, Rightl), for all possible pairs of populations in the proposed “Left” and “Right sets. It then determines whether all the statistics can be written as a linear combination of f4-statistics corresponding to the differentiation patterns between the proposed N − 1 ancestral populations, appropriately accounting for the covariance of these statistics and computing a single p value for fit based on a Hotelling T-squared distribution36. For models that are consistent with the data (p > 0.05), qpAdm estimates proportions of admixture for the target population from sources related to the N − 1 ancestral populations (with standard errors). Crucially, qpAdm does not require specifying an explicit model for how the “Right” outgroup populations are related.

We first examined all possible “Left” population sets that consisted of Levant_ChL along with one other ancient population from the analysis dataset. Testing a wide range of ancient populations, we found that p values for all possible Left populations were below 0.05 (Supplementary Data 2), showing that Levant_ChL is not consistent with being a clade with any of them relative to the “Right” 09NW outgroups. We then considered models with “Left” population sets containing Levant_ChL along with two additional ancient populations, which corresponds to modeling the Levant_ChL as the result of a two-way admixture between populations related to these two other ancient populations. To reduce the number of hypotheses tested, we restricted the models to pairs of source populations that contain at least one of the six populations that we consider to be the most likely admixture sources based on geographical and temporal proximity: Anatolia_N, Anatolia_ChL, Armenia_ChL, Iran_ChL, Iran_N, and Levant_N. Again, we find no plausible two-way admixture models using a p > 0.05 threshold (Supplementary Figure 2 and Supplementary Data 3). Finally, we tested possible three-way admixture events, restricting to triplets that contain at least two of the six most likely admixture sources. Plausible solutions at p > 0.05 are listed in Table 1 (full results are reported in Supplementary Figure 3 and Supplementary Data 4).

We found multiple candidates for three-way admixture models, always including (1) Levant_N (2) either Anatolia_N or Europe_EN and (3) either Iran_ChL, Iran_N, Iran_LN, Iran_HotuIIIb or Levant_BA_North. These are all very similar models, as Europe_EN (early European agriculturalists) are known to be genetically primarily derived from Anatolian agriculturalists (Anatolia_N)31, and Levant_BA_North has ancestry related to Levant_N and Iran_ChL26. To distinguish between models involving Anatolian Neolithic (Anatolia_N) and European Early Neolithic (Europe_EN), we repeated the analysis including additional outgroup populations in the “Right” set that are sensitive to the European hunter-gatherer-related admixture present to a greater extent in Europe_EN than in Anatolia_N (Supplementary Figure 4a)31 (thus, we added Switzerland_HG, SHG, EHG, Iberia_BA, Steppe_Eneolithic, Europe_MNChL, Europe_LNBA to the “Right” outgroups; abbreviations in Supplementary Table 2). We found that only models involving Levant_N, Anatolia_N, and either Iran_ChL or Levant_BA_North passed at p > 0.05 (Table 1). To distinguish between Iran_ChL and Levant_BA_North, we added Iran_N to the outgroup set (for a total of 19 = 11 + 8 outgroups) (Supplementary Figure 4b). Only the model involving Iran_ChL remained plausible. Based on this uniquely fitting qpAdm model we infer the ancestry of Levant_ChL to be the result of a three-way admixture of populations related to Levant_N (57%), Iran_ChL (17%), and Anatolia_N (26%).

### The ancestry of late Levantine Bronze Age populations

It was striking to us that previously published Bronze Age Levantine samples from the sites of 'Ain Ghazal in present-day Jordan (Levant_BA_South) and Sidon in present-day Lebanon (Levant_BA_North) can be modeled as two-way admixtures, without the Anatolia_N contribution that is required to model the Levant_ChL population24,26. This suggests that the Levant_ChL population may not be directly ancestral to these later Bronze Age Levantine populations, because if it were, we would also expect to detect an Anatolia_N component of ancestry. In what follows, we treat Levant_BA_South and Levant_BA_North as separate populations for analysis, since the symmetry statistic f4(Levant_BA_North, Levant_BA_South; A, Chimp) is significant for a number test populations A (|Z| ≥ 3) (Supplementary Data 5), consistent with the different estimated proportions of Levant_N and Iran_ChL ancestry reported in24,26.

To test the hypothesis that Levant_ChL may be directly ancestral to the Bronze Age Levantine populations, we attempted to model both Levant_BA_South and Levant_BA_North as two-way admixtures between Levant_ChL and every other ancient population in our dataset, using the base 09NW set of populations as the “Right” outgroups. We also compared these models to the previously published models that used the Levant_N and Iran_ChL populations as sources (Table 2; Supplementary Figure 5; Supplementary Data 6). In the case of Levant_BA_South from 'Ain Ghazal, Jordan, multiple models were plausible, and thus we returned to the strategy of adding additional “Right” population outgroups that are differentially related to one or more of the “Left” populations (specifically, we added various combinations of Armenia_EBA, Steppe_EMBA, Switzerland_HG, Iran_LN, and Iran_N). Only the model including Levant_N and Iran_ChL remains plausible under all conditions. Thus, we can conclude that groups related to Levant_ChL contributed little ancestry to Levant_BA_South.

We observe a qualitatively different pattern in the Levant_BA_North samples from Sidon, Lebanon, where models including Levant_ChL paired with either Iran_N, Iran_LN, or Iran_HotuIIIb populations appear to be a significantly better fit than those including Levant_N + Iran_ChL. We largely confirm this result using the “Right” population outgroups defined in Haber et al.26 (abb. Haber: Ust_Ishim, Kostenki14, MA1, Han, Papuan, Ami, Chuckhi, Karitiana, Mbuti, Switzerland_HG, EHG, WHG, and CHG), although we find that the specific model involving Iran_HotuIIIb no longer works with this “Right” set of populations. Investigating this further, we find that the addition of Anatolia_N in the “Right” outgroup set excludes the model of Levant_N + Iran_ChL favored by Haber et al.26. These results imply that a population that harbored ancestry more closely related to Levant_ChL than to Levant_N contributed to the Levant_BA_North population, even if it did not contribute detectably to the Levant_BA_South population.

We obtained additional insight by running qpAdm with Levant_BA_South as a target of two-way admixture between Levant_N and Iran_ChL, but now adding Levant_ChL and Anatolia_N to the basic 09NW “Right set of 11 outgroups. The addition of the Levant_ChL causes the model to fail, indicating that Levant_BA_South and Levant_ChL share ancestry following the separation of both of them from the ancestors of Levant_N and Iran_ChL. Thus, in the past there existed an unsampled population that contributed both to Levant_ChL and to Levant_BA_South, even though Levant_ChL cannot be the direct ancestor of Levant_BA_South because, as described above, it harbors Anatolia_N-related ancestry not present in Levant_BA_South.

### Genetic heterogeneity in the Levantine Bronze Age

We were concerned that our finding that the Levant_ChL population was a mixture of at least three groups might be an artifact of not having access to samples closely related to the true ancestral populations. One specific possibility we considered is that a single ancestral population admixed into the Levant to contribute to both the Levant_ChL and the Levant_BA_South populations, and that this was an unsampled population on an admixture cline between Anatolia_N and Iran_ChL, explaining why qpAdm requires three source populations to model it. To formally test this hypothesis, we used qpWave36,37,38, which determines the minimum number of source populations required to model the relationship between “Left populations relative to “Right outgroup populations. Unlike qpAdm, qpWave does not require that populations closely related to the true source populations are available for analysis. Instead it treats all “Left” populations equally, and attempts to determine the minimum number of theoretical source populations required to model the “Left” population set, relative to the “Right” population outgroups. Therefore, we model the relationship between Levant_N, Levant_ChL, and Levant_BA_South as “Left populations, relative to the 09NW “Right” outgroup populations (Table 3). We find that a minimum of three source populations continues to be required to model the ancestry of these Levantine populations, supporting a model in which at least three separate sources of ancestry are present in the Levant between the Neolithic, Chalcolithic, and Bronze Age.

We applied qpWave again, replacing Levant_ChL with Levant_BA_North, and found that the minimum number of source populations is only two. However, when we include the Levant_ChL population as an additional outgroup, three source populations are again required. This suggests that in the absence of the data from Levant_ChL there is insufficient statistical leverage to detect Anatolian-related ancestry that is truly present in admixed form in the Levant_BA_North population (data from the Levant_ChL population makes it possible to detect this ancestry). This may explain why Haber et al.26 did not detect the Anatolian Neolithic-related admixture in Levant_BA_North.

### Biologically important mutations in the Peqi’in population

This study nearly doubles the number of individuals with genome-wide data from the ancient Levant. Measured in terms of the average coverage at SNPs, the increase is even more pronounced due to the higher quality of the data reported here than in previous studies of ancient Near Easterners24,26. Thus, the present study substantially increases the power to analyze the change in frequencies of alleles known to be biologically important.

We leveraged our data to examine the change in frequency of SNP alleles known to be related to metabolism, pigmentation, disease susceptibility, immunity, and inflammation in the Levant_ChL population, considered in relation to allele frequencies in the Levant_N, Levant_BA_North, Levant_BA_South, Anatolia_N and Iran_ChL populations and present-day pools of African (AFR), East Asian (EAS), and European (EUR) ancestry in the 1000 Genomes Project Phase 3 dataset39 (Supplementary Data 7).

We highlight three findings of interest. First, an allele (G) at rs12913832 near the OCA2 gene, with a proven association to blue eye color in individuals of European descent40, has an estimated alternative allele frequency of 49% in the Levant_ChL population, suggesting that the blue-eyed phenotype was common in the Levant_ChL.

Second, an allele at rs1426654 in the SLC24A5 gene which is one of the most important determinants of light pigmentation in West Eurasians41 is fixed for the derived allele (A) in the Levant_ChL population suggesting that a light skinned phenotype may have been common in this population, although any inferences about skin pigmentation based on allele frequencies observed at a single site need to be viewed with caution42.

Third, an allele (G) at rs6903823 in the ZKSCAN3 and ZSCAN31 genes which is absent in all early agriculturalists reported to date (Levant_N, Anatolia_N, Iran_N) and that has been argued to have been under positive selection by Mathieson et al.31, occurs with an estimated frequency of 20% in the Levant_ChL, 17% in the Levant_BA_South, and 15% in the Iran_ChL populations, while it is absent in all other populations. This suggests that the allele was rising in frequency in Chalcolithic and Bronze Age Near Eastern populations at the same time as it was rising in frequency in Europe.

## Discussion

The Chalcolithic period in the Levant witnessed major cultural transformations in virtually all areas of culture, including craft production, mortuary and ritual practices, settlement patterns, and iconographic and symbolic expression43. The current study provides insight into a long-standing debate in the prehistory of the Levant, implying that the emergence of the Chalcolithic material culture was associated with population movement and turnover.

The quality of ancient DNA obtained from the Peqi’in Cave samples is excellent relative to other sites in the Near East. We hypothesize that the exceptional preservation is due to two factors. First, the targeted sampling of ancient DNA from the petrous portion of the temporal bone makes it possible to obtain high-quality ancient DNA from previously inaccessible geographic regions24,27,44,45. Secondly, the environment of Peqi’in Cave is likely to be favorable for DNA preservation. The skeletal remains—either stored in ossuaries or laid in the ground—were quickly covered by a limestone crust, isolating them from their immediate surroundings and protecting them from acidic conditions that are known to be damaging to DNA.

We find that the individuals buried in Peqi’in Cave represent a relatively genetically homogenous population. This homogeneity is evident not only in the genome-wide analyses but also in the fact that most of the male individuals (nine out of ten) belong to the Y-chromosome haplogroup T (see Supplementary Table 1), a lineage thought to have diversified in the Near East46. This finding contrasts with both earlier (Neolithic and Epipaleolithic) Levantine populations, which were dominated by haplogroup E24, and later Bronze Age individuals, all of whom belonged to haplogroup J24,26.

Our finding that the Levant_ChL population can be well-modeled as a three-way admixture between Levant_N (57%), Anatolia_N (26%), and Iran_ChL (17%), while the Levant_BA_South can be modeled as a mixture of Levant_N (58%) and Iran_ChL (42%), but has little if any additional Anatolia_N-related ancestry,  can only be explained by multiple episodes of population movement. The presence of Iran_ChL-related ancestry in both populations – but not in the earlier Levant_N – suggests a history of spread into the Levant of peoples related to Iranian agriculturalists, which must have occurred at least by the time of the Chalcolithic. The Anatolian_N component present in the Levant_ChL but not in the Levant_BA_South sample suggests that there was also a separate spread of Anatolian-related people into the region. The Levant_BA_South population may thus represent a remnant of a population that formed after an initial spread of Iran_ChL-related ancestry into the Levant that was not affected by the spread of an Anatolia_N-related population, or perhaps a reintroduction of a population without Anatolia_N-related ancestry to the region. We additionally find that the Levant_ChL population does not serve as a likely source of the Levantine-related ancestry in present-day East African populations (see Supplementary Note 4)24.

These genetic results have striking correlates to material culture changes in the archaeological record. The archaeological finds at Peqi’in Cave share distinctive characteristics with other Chalcolithic sites, both to the north and south, including secondary burial in ossuaries with iconographic and geometric designs. It has been suggested that some Late Chalcolithic burial customs, artifacts and motifs may have had their origin in earlier Neolithic traditions in Anatolia and northern Mesopotamia8,13,47. Some of the artistic expressions have been related to finds and ideas and to later religious concepts such as the gods Inanna and Dumuzi from these more northern regions6,8,47,48,49,50. The knowledge and resources required to produce metallurgical artifacts in the Levant have also been hypothesized to come from the north11,51.

Our finding of genetic discontinuity between the Chalcolithic and Early Bronze Age periods also resonates with aspects of the archeological record marked by dramatic changes in settlement patterns43, large-scale abandonment of sites52,53,54,55, many fewer items with symbolic meaning, and shifts in burial practices, including the disappearance of secondary burial in ossuaries56,57,58,59. This supports the view that profound cultural upheaval, leading to the extinction of populations, was associated with the collapse of the Chalcolithic culture in this region18,60,61,62,63,64.

These ancient DNA results reveal a relatively genetically homogeneous population in Peqi’in. We show that the movements of people within the region of the southern Levant were remarkably dynamic, with some populations, such as the one buried at Peqi’in, being formed in part by exogenous influences. This study also provides a case-study relevant beyond the Levant, showing how combined analysis of genetic and archaeological data can provide rich information about the mechanism of change in past societies.

## Methods

### Data generation

We screened all libraries for authentic DNA by enriching for the mitochondrial genome and 50 nuclear target loci, followed by sequencing on an Illumina NextSeq500 instrument for 2 × 75 cycles and 2 × 7 cycles to read out the indices. We enriched promising libraries for approximately 1.2 M SNPs as described in refs. 31,36,66,] and then sequenced on a NextSeq500 sequencer using 75 base pair paired-end sequences. During computational processing, we initially stripped identifying oligonucleotide sequences and adapters, separating individual samples from pooled captures by their identifying 7 base pair indices at the 5′ and 3′ ends of reads, and requiring matches to sample-specific barcodes appended directly to the sequence fragments, allowing no more than one mismatch per index/barcode. We used SeqPrep67 to strip adapters and also to merge paired end reads into single sequences by requiring a minimum of 15 base pair overlap (allowing up to one mismatch), using the highest quality base in the merged region where there was a conflict. We used samse in bwa (v0.6.1)68 to align reads. For the mitochondrial DNA enrichment experiment we aligned to the RSRS mitochondrial genome69. For the whole-genome enrichment experiment we aligned to the hg19 reference genome. We identified duplicate sequences as ones with the same start and end positions and orientation and also identical barcode pairs, and retained the highest quality sequence from each duplicate. We made pseudo-haploid SNP calls for each position using a randomly chosen sequence covering each targeted site, stripping the two bases at the ends of each sequence to remove deaminated mutations, and requiring a minimum mapping quality (MAPQ ≥ 10), and restricting to sites with a minimum base quality (≥20).

We assessed the quality of each library at the screening stage using three standard methods for determining ancient DNA authenticity. First, we analyzed mitochondrial genome data to determine the rate of matching to the consensus sequence, using contamMix30. Second, we restricted to samples in which the rate of C-to-T substitutions in terminal nucleotides was at least 3%, as expected for genuine ancient DNA using the partial UDG treatment protocol29. Finally, we used the ANGSD software to obtain a conservative estimate of contamination in the X-chromosome of individuals determined to be male based on the rate of polymorphism on X-chromosome sequences (males have only a single X-chromosome and so are not expected to show polymorphism); we excluded libraries with X-contamination estimates greater than 1.5%32. For samples where multiple libraries were produced for a single individual, we merged libraries that passed quality control, and obtained new pseudo-haploid SNP calls.

We determined mitochondrial DNA haplogroups using the tool haplogrep270, using a consensus sequence built from reads enriched for the mitochondrial genome, restricting to damaged reads using PMDtools71 (pmdscore ≥ 3), and trimming 5 bases from each end to greatly reduce the error rate due to deamination.

Ancient DNA presents challenges in the assignment of Y-chromosome haplogroups due to the chance that there may be contamination, DNA damage or missing data present in them. In order to assign Y haplogroups to our data, we used a modified version of the procedure used in the analysis of modern Y chromosomes in the 1000 Genomes Project72, which uses a breadth-first search to traverse the Y-chromosome tree. We made our calls on the ISOGG tree from 04.01.2016 [http://isogg.org], and modified the caller to output derived and ancestral allele calls for each informative position on the tree. We then assigned a score to each of the reference haplogroups by counting the number of mismatches in the number of observed derived alleles on that branch and down-weighted derived mutations that were transitions to 1/3 that of transversions to account for DNA damage related errors. We assigned the sample to the reference haplogroup with the closest match based on this score. While we endeavored to produce a call on each sample, we note that samples with fewer than 100,000 SNPs have too little data to confidently identify the correct haplogroup, and we encourage caution when interpreting these results.

The data from the 22 samples that passed contamination and quality control tests are reported in Supplementary Table 1, with an average of 0.97× coverage on the 1240 k SNP targets, and an average of 358,313 SNPs covered at least once. A by-library table describing the screening results is reported in Supplementary Data 1. We excluded two individuals from further analysis, as the genetic patterns observed using the method described in Kuhn et al.73. Showed that they were first-degree relatives of higher coverage samples in the dataset. We restricted data from sample I1183 to include only sequences with evidence of C-to-T substitution in order to minimize contamination which was evident in the full data from these samples.

We combined the newly reported data with existing data from Lazaridis et al.24 and Haber et al.26, using the mergeit program of EIGENSOFT33. The resulting datasets, referred to as HO + and HOIll+, contain the 20 new unrelated samples combined with HO and HOIll from Lazaridis et al.24 and 5 ancient samples from Sidon, Lebanon (population name: Levant_BA_North) from Haber et al.26, respectively. HO+ includes data from 2891 modern and ancient individuals at 591,642 SNPs, and HOIll+ includes data from 306 ancient individuals at 1,054,637 SNPs.

### Principal component analysis

We performed PCA on the HO+ dataset using smartpca33. We used a total of 984 present-day individuals for PCA, and projected the 306 ancient samples. We used default parameters with lsqproject: YES and numoutlieriter: 0 settings. We estimated FST using smartpca for the 21 ancient Near Eastern populations made up of more than one individual and 8 modern populations using default parameters, with inbreed: YES and fstonly: YES (Supplementary Figure 1). We ran analyses using the HO+ dataset.

We carried out ADMIXTURE analysis34 on the HO+ dataset. Prior to analyses, we pruned SNPs in strong linkage disequilibrium with each other using PLINK74 using the parameters—indep-pairwise 200 25 0.4. We performed ADMIXTURE analysis on the 3,00,885 SNPs remaining in the pruned dataset. For each value of k between 2 and 14, we performed 20 replicate analyses, and we retained the highest likelihood replicate for each k.

### Conditional heterozygosity

We computed conditional heterozygosity for each ancient Levantine population using popstats75. For this analysis we used the HO+ dataset, restricting to SNP sites ascertained from a single Yoruba individual and to transversion SNPs, as described in Skoglund et al.44.

### f-statistics

We computed f4-statistics using the qpDstat program in ADMIXTOOLS35, with default parameters, and f4 mode:YES. We computed f3-statistics using the qp3Pop program in ADMIXTOOLS35, using default parameters, with inbreed: YES. We ran all analyses using the HOIll+ dataset, except for the statistic f4(Levant_BA_North, Levant_BA_South; A, Chimp), which we ran on the HO+ dataset.

We estimated proportions of ancestry in the Levant_ChL population using the qpAdm methodology, with parameters allsnps: YES and details:YES36. We tested both 2- and 3-way admixtures between ancient “Left” populations from the HOIll+ dataset. We used the 09NW populations defined in Lazaridis et al.24 as preliminary outgroups. We selected additional outgroups based on the statistics f4(Anatolia_N, Europe_EN; A, Chimpanzee) and f4(Levant_BA_North, Iran_ChL; A, Chimpanzee), and we repeated qpAdm with each additional outgroup added into the “Right” list until all but one admixture model was eliminated.

We used qpAdm to determine whether the Levant_BA_South and Levant_BA_North populations could be modeled using Levant_ChL as a source population. We tested 2-way admixtures between Levant_ChL and every other ancient “Left” population from the HOIll+ dataset. We also tested the “Left” populations Levant_N and Iran_ChL. We used the 09NW “Right populations as preliminary outgroup populations, and confirmed our findings for Levant_BA_North using the outgroups defined in Haber et al.26. We added additional outgroups to further differentiate between plausible models, and repeated qpAdm analysis until all but one candidate admixture model was eliminated.

### qpWave

We computed the minimum number of streams of ancestry required to model two sets of three Levantine populations (set [1] Levant_N, Levant_ChL, and Levant_BA_South, set [2] Levant_N, Levant_BA_South, Levant_BA_North) using the qpWave37,38 methodology with parameter allsnps:YES.

### Allele frequency comparisons

We examined the frequencies of SNPs associated with phenotypically important functions in the categories of metabolism, pigmentation, disease susceptibility, immunity, and inflammation in Levant_ChL in conjunction with the Levant_N, Levant_BA_North, Levant_BA_South, Anatolia_N and Iran_ChL populations, with allele frequencies for three pooled continental populations (AFR, EAS, EUR) in Phase 3 the 1000 Genomes Project reported where available. We computed allele frequencies at each site of interest by computing the likelihood of the population reference allele frequency given the data, using a method established in Mathieson et al.31. For each population of size, N, we observe Ri sequences that possess the reference allele out of a total Ti sequences. The likelihood of the reference allele frequency, p, in each population given the data D = {X,N,Ri,Ti} is L(p;D) = $$\mathop {\prod }\limits_{i = 1}^N \{ p^2B\left( {R_i,T_i,1 - \varepsilon } \right) + 2p\left( {1 - p} \right)B\left( {R_i,T_i,0.5} \right) + (1 - p)^2B\left( {R_i,T_i,\varepsilon } \right)\}$$ where B(k,n,p) = $$\left( {\begin{array}{*{20}{c}} n \\ k \end{array}} \right)p^k(1 - p)^{n - k}$$ is the binomial probability distribution, and $$\varepsilon$$ is a small probability of error, which we set to 0.001 for our calculations. We estimated allele frequencies by maximizing the likelihood numerically for each population.

### Data availability

The aligned sequences are available through the European Nucleotide Archive under accession number PRJEB27215. Genotype datasets used in analysis are available at https://reich.hms.harvard.edu/datasets.

Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Change history

• ### 05 September 2018

This Article was originally published without the accompanying Peer Review File. This file is now available in the HTML version of the Article; the PDF was correct from the time of publication.

• ### 20 September 2018

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## Acknowledgments

Peqi’in burial Cave was excavated under the auspices of the Israel Antiquities Authority. E.H. was supported by a graduate student fellowship from the Max Planck–Harvard Research Center for the Archaeoscience of the Ancient Mediterranean (MHAAM). D.R. was supported by the U.S. National Science Foundation HOMINID grant BCS-1032255, the U.S. National Institutes of Health grant GM100233, by an Allen Discovery Center grant, and is an investigator of the Howard Hughes Medical Institute. The anthropological study was supported by the Dan David Foundation. We thank Vagheesh Narasimhan for generating and describing Y-chromosome haplogroup calls. We thank Ariel Pokhojaev for creating the map image used in Fig. 1a. We thank John Wakeley for critical comments.

## Author information

### Author notes

1. These authors contributed equally: Éadaoin Harney, Hila May.

2. These authors jointly supervised this work: Israel Hershkovitz, David Reich.

### Affiliations

2. #### Department of Genetics, Harvard Medical School, Boston, MA, 02115, USA

• , Swapan Mallick
• , Iosif Lazaridis
• , Kristin Stewardson
• , Susanne Nordenfelt
•  & David Reich
3. #### The Max Planck–Harvard Research Center for the Archaeoscience of the Ancient Mediterranean, Cambridge, MA, 02138, USA

• , Iosif Lazaridis
•  & David Reich
4. #### Department of Anatomy and Anthropology, Sackler Faculty of Medicine, Tel Aviv University, Tel Aviv, 6997801, Israel

• Hila May
•  & Israel Hershkovitz
5. #### Shmunis Family Anthropology Institute, Dan David Center for Human Evolution and Biohistory Research, Sackler Faculty of Medicine, Steinhardt Natural History Museum, Tel Aviv University, Tel Aviv, 6997801, Israel

• Hila May
• , Rachel Sarig
•  & Israel Hershkovitz
6. #### The Institute for Galilean Archaeology, Kinneret Academic College, Kinneret, 15132, Israel

• Dina Shalem
7. #### Broad Institute of MIT and Harvard, Cambridge, 02142, MA, USA

• Swapan Mallick
• , Nick Patterson
•  & David Reich
8. #### Howard Hughes Medical Institute, Boston, MA, 02115, USA

• Swapan Mallick
• , Kristin Stewardson
• , Susanne Nordenfelt
• , Nick Patterson
•  & David Reich
9. #### The Maurice and Gabriela Goldschleger School of Dental Medicine, Sackler Faculty of Medicine, Tel Aviv University, Tel Aviv, 6997801, Israel

• Rachel Sarig

### Contributions

H.M, I.H., and D.R. conceived the study. D.R. supervised the ancient DNA work, sequencing, and data analysis. H.M, D.S, R.S., and I.H. assembled, studied, or described the archaeological material. E.H., H.M., N.R., K.S, and S.N performed or supervised wet laboratory work. S.M performed bioinformatics analyses. E.H. performed population genetics analyses, with I.L. and N.P. providing guidance. E.H., H.M, I.H, and D.R wrote the manuscript with input from all co-authors.

### Competing interests

The authors declare no competing interests.

### Corresponding authors

Correspondence to Éadaoin Harney or Hila May.


text:
0
TECHNICAL PAPERS: Gas Turbines: CFD Modeling and Simulation

# Rayleigh-Bénard Convection in Open and Closed Rotating Cavities

[+] Author and Article Information
Martin P. King

Abdus Salam International Centre for Theoretical Physics, Strada Costiera 11, Trieste, Italy

Michael Wilson

Department of Mechanical Engineering, University of Bath, Bath BA2 7AY, UK

J. Michael Owen

Department of Mechanical Engineering, University of Bath, Bath BA2 7AY, UKensjmo@bath.ac.uk

J. Eng. Gas Turbines Power 129(2), 305-311 (Nov 15, 2005) (7 pages) doi:10.1115/1.2432898 History: Received October 31, 2005; Revised November 15, 2005

## Abstract

Buoyancy effects can be significant in the rotating annular cavities found between compressor discs in gas-turbine engines, where Rayleigh numbers above $1012$ are common. In some engines, the cavity is “closed” so that the air is confined between four rotating surfaces: two discs and inner and outer cylinders. In most engines, however, the cavity is “open” and there is an axial throughflow of cooling air at the center. For open rotating cavities, a review of the published evidence suggests a Rayleigh–Bénard type of flow structure, in which, at the larger radii, there are pairs of cyclonic and anti-cyclonic vortices. The toroidal circulation created by the axial throughflow is usually restricted to the smaller radii in the cavity. For a closed rotating annulus, solution of the unsteady Navier–Stokes equations, for Rayleigh numbers up to $109$, show Rayleigh–Bénard convection similar to that found in stationary enclosures. The computed streamlines in the $r$-$θ$ plane show pairs of cyclonic and anti-cyclonic vortices; but, at the larger Rayleigh numbers, the computed isotherms suggest that the flow in the annulus is thermally mixed. At the higher Rayleigh numbers, the computed instantaneous Nusselt numbers are unsteady and tend to oscillate with time. The computed time-averaged Nusselt numbers are in good agreement with the correlations for Rayleigh–Bénard convection in a stationary enclosure, but they are significantly higher than the published empirical correlations for a closed rotating annulus.

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## Figures

Figure 1

Simplified diagram of high-pressure compressor rotor with axial throughflow

Figure 2

Schematic diagram of open and closed rotating cavities

Figure 3

Rayleigh–Bénard convection in a horizontal stationary enclosure

Figure 4

Computed 2D streamlines and isotherms for a∕b=0.5

Figure 5

Computed 3D flow structure at z¯=0.25 and for Ra3=103.9, a∕b=0.5, G=0.5

Figure 6

2D computations of variation of Nu3,av with t¯ for a∕b=0.5

Figure 7

2D computations of variation of Nu¯3,av with Ra3 for a∕b=0.5

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text:
# Gravitational waves speed in a medium

1. Feb 17, 2016

### Gerinski

Electromagnetic waves are known to travel a c in a vacuum, but at lower speeds in a material medium.
What about gravitational waves? They are also predicted to travel at c in a vacuum, but what about them traveling through material mediums? Do they get slowed down? by which factor?

TX

2. Feb 17, 2016

### andrewkirk

I doubt there has been much work done on that. The calculations I have seen regarding gravitational waves have all assumed that outside the source region there is negligible mass-energy, and accordingly assumes the stress-energy tensor outside the source to be zero. Since a medium has mass, the question of speed through a medium would be outside the scope of such calculations.

3. Feb 17, 2016

### bcrowell

Staff Emeritus
See Ingraham, "Gravitational waves in matter," GRG 29 (1997) 117. Ingraham suggests that the best chance for seeing $v<c$ might be in "gravitational wave propagation through molecular gas clouds of galactic or intergalactic size." Although n−1 would be incredibly small, he suggests that you might be able to see an effect accumulated over thousands or millions of light-years.

Note that the recent aLIGO result puts a very tight upper limit on dispersion.


text:
# Why are we so careful in saying that dy/dx is not a fraction?

Calculus instructors are mostly very careful to explain that $\dfrac{dy}{dx}$ is not a fraction, and multiplying both sides of an equation by dx is nonsense, wrong, or evil.

However, I really want to be able to say that "y is the sum of all the little pieces of y," i.e. $y = \int dy$.

And I'd really like to say that when I integrate $\dfrac{dy}{dx} dx$, it's the same as integrating $dy$.

And when I separate variables, I really like splitting up the $\dfrac{dy}{dx}$.

This comes to the question: what is a good example of the harm that we will cause if we allow students to think $\dfrac{dy}{dx}$ is a fraction? Are we worried that students will think that $dy$ and $dx$ are numbers?

-
This question has been asked before on MO: mathoverflow.net/q/73492/22971 ("How misleading is it to regard dy/dx as a fraction?") – Benjamin Dickman Aug 28 '14 at 2:03
(Though the answers at a similar MSE query linked from MO - math.stackexchange.com/q/21199/37122 - may prove to be more useful.) – Benjamin Dickman Aug 28 '14 at 2:08
There's no reason why you can't think of dx and dy as one forms on xy space. You can't divide one forms but if you have a relation like dy = 2xdx then you can think of that as picking out a one-dimensional subspace defined by the one form dy - 2xdx. On that subspace dy and dx are still one forms but indistinguishable from scalars so you can form their ratio. This doesn't work with partials because those aren't ratios of one-forms but inner products of one forms with basis vectors. – Francis Davey Aug 28 '14 at 9:44
Can't you say that integrating $\dfrac{dy}{dx} dx$ is the same as integrating $dy$ "because there's a theorem that says so", rather than saying that integrating $\dfrac{dy}{dx} dx$ is the same as integrating $dy$ "because $\dfrac{dy}{dx} dx$ is a fraction and the $dx$s cancel"? – Steve Jessop Aug 28 '14 at 12:46
Just a small note that people have tried to rigorize the use of $dy$ and $dx$ as infinitesimals, calling it non-standard analysis. I don't really know anything about it, but I guess it's at least a disclaimer that there is a whole other Wikipedia article about criticism of non-standard analysis. – MGA Aug 28 '14 at 15:14

If $y$ is a highly oscillating function of $x$, it can become quite unclear exactly what is going on with domains of integration if one tries to write out in precise terms the kind of "integration identity" you would like to express (in terms of definite integration). My high school physics teacher used exactly the kind of reasoning your are suggesting (with integrating $(dy/dx)dx = dy$ to get $y$) as a way of suggesting that the Fundamental Theorem of Calculus is almost a triviality, or a matter of defining notation (without carefully thinking through why the notation is well-posed, etc.).

Since we know that the Fundamental Theorem of Calculus is a real theorem, the danger of putting too much emphasis on the fraction notation is that it deludes students into thinking that FTC has no content, or is some kind of triviality. For most students the goal can't be to make them appreciate the subtleties of definitions and proofs of theorems, but it would be good to at least make sure they understand that the fractional formalism gives "right answers" because it is well-designed notation masking the content of some real theorems (a fact which is hard for them to accept if they think the derivatives literally are fractions). That isn't to say one should completely divorce the intuition behind the notation from the properties one wants it to satisfy, of course.

Overall, the fractional notation does work really well (for Chain Rule, u-substitution, derivative of inverse functions, etc.), so I wouldn't belabor the point after making one attempt with an example of non-invertible $y$ that the possibility to compute "as if" $y$ were a variable in its own right is an amazing feature of Leibniz' notation.

That being said, I think the situation with the multivariable chain rule mentioned in one of the other answers is a very good illustration of why too much blind faith in the fractional notation without carefully thinking through why it is working will lead students into traps when they move on to multivariable calculus (and the reality is that most students cannot be expected to think very carefully about the notation when they haven't mastered or been taught the precise definitions).

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The other answers are also very good, but I really think the earliest key point is that integrating $\dfrac{dy}{dx}dx$ to get $y$ is not a triviality (it's a rather important theorem!), so I'm accepting this one. – Chris Cunningham Aug 29 '14 at 17:09
@ChrisCunningham I think it is important to realize that the idea of the fundamental theorem of calculus is kind of trivial. A formal proof might be a bit tricky, but it is fundamentally a very intuitive notion. Worth making sure students realize that. – Steven Gubkin Aug 30 '14 at 2:10
@StevenGubkin: The idea of the proof of Serre's GAGA theorem comparing complex-analytic and complex-algebraic geometry is also kind of trivial...as is the idea of the proof of effectivity of faithfully flat descent for morphisms of schemes...and so are the ideas of the proofs of lots of other deep results once someone first has the idea! It does not help students new to a mathematical topic to tell them that the basic idea behind a key theorem is "kind of trivial". Hindsight is 20-20, but foresight is not. I believe that your advice, while well-intentioned, will not help students. – user52824 Aug 30 '14 at 3:21
@user52824 I think most students (at least at my school) do not have even an intuitive understanding of FTC. Conveying some kind of intuition about FTC is important. If this intuition is "$\frac{dy}{dx}$ is the little change in $y$ divided by the little change in $x$, so $\frac{dy}{dx}dx$ is the little change in $y$, so if I add all of these up I get the change in $y$ over the whole interval", I think that is great. They should understand it is not a formal proof, but it is still better than just memorizing a formula. – Steven Gubkin Aug 30 '14 at 15:10
@StevenGubkin: Sure, I agree with you that imparting some intuition as to why FTC should be true is an important part of teaching calculus. But I would steer very clear of using the phrase "kind of trivial" to describe a result whose full significance is hard to grasp by most students without a lot of effort (i.e., most students do not consider calculus to be an easy subject to learn); that is all I was trying to convey in my response to your comment. – user52824 Aug 30 '14 at 23:15

A big issue I've seen is in understanding the multivariable chain rule. I've had many students over the years argue that $$\frac{\partial f}{\partial t} = \frac{\partial f}{\partial x}\frac{\partial x}{\partial t}+\frac{\partial f}{\partial y}\frac{\partial y}{\partial t}$$ implies $$\frac{\partial f}{\partial t} = 2\frac{\partial f}{\partial t}, \text{ so } \frac{\partial f}{\partial t}=0$$ since the $\partial x$ and $\partial y$ cancel.

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The problem here isn't that the student is reasoning using infinitesimals. You can make up an example with $f(x,y)$ a bilinear function, and $x(t)$ and $y(t)$ linear functions, convert all the $\partial$s to $\Delta$s, and the mistake in the calculation is the same mistake, even though everything is now a real number and no calculus is involved. The mistake has nothing to do with calculus. The mistake is a logical mistake about not understanding that one variable is kept constant while another one is varied. – Ben Crowell Aug 28 '14 at 3:16
Indeed the problem isn't that the student is reasoning using infinitesimals. The student in fact isn't doing that. The problem is that the student is treating the partial derivative notation as a fraction, which is what was asked for except that "partial" isn't what was in the question. – Steve Jessop Aug 28 '14 at 12:36
I am highly sympathetic to this answer (and it's the first one that sprang to mind), but I think that there may be an important subtlety: as Ben Crowell points out below (matheducators.stackexchange.com/a/4288/2070), there is a rigorous sense in which $\text dy/\text dx$ really is a fraction, so that no logical contradiction can result from treating it as such. (I think that there is no rigorous sense in which $\partial z/\partial x$ and $\partial z/\partial y$ are simultaneously fractions; but do not know enough of infinitesimals to be confident saying so.) – L Spice Aug 29 '14 at 23:28
(The punnish confusion between being "really a fraction", i.e., genuinely such an object, but not "a real fraction", i.e., a fraction of real numbers, was inadvertent.) – L Spice Aug 29 '14 at 23:29

When Leibniz invented the notation, he considered $dx$ and $dy$ to be infinitesimal numbers and $dy/dx$ to be their quotient. Non-standard analysis (NSA) has essentially vindicated that view. One has to distinguish between the quotient $dy/dx$ and its standard part, which is the derivative. In fact, Leibniz made this distinction [Blaszczyk 2012], although it was not carefully observed by later users of his notation.

Keisler is a very nice, free freshman calc book that develops calculus from the NSA point of view.

There was a period ca. 1820-1965 when this stuff was not clearly understood, and it was believed, incorrectly, that limits were the only possible way to give calculus a rigorous foundation. Even during this period, scientists and engineers never stopped using infinitesimals. Freshman calc students need to be literate in the practices of science and engineering, so a disservice is done to them when the discredited belief is inculcated upon them that these practices are somehow wrong, ignorant, or logically suspect.

This comes to the question: what is a good example of the harm that we will cause if we allow students to think $\dfrac{dy}{dx}$ is a fraction? Are we worried that students will think that $dy$ and $dx$ are numbers?

There is no such harm. Students should of course be told that $dx$ and $dy$ are not real numbers. In old-school terminology, they're infinitesimal numbers. In the termninology of NSA, they're hyperreal numbers.

Blaszczyk et al., "Ten Misconceptions from the History of Analysis and Their Debunking," http://arxiv.org/abs/1202.4153

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Indeed. And it is still very widely believed, incorrectly, that only the Cauchy-Weierstrass approach can legitimize calculus. And caricatures of Leibniz (and Euler's) viewpoints on infinitesimals are still widely taken as accurate. Berkeley's objections had substance, but the whole thing was inconclusive even at the time. – paul garrett Aug 28 '14 at 19:11

On the same line of Santiago Canez's answer, an identity that appears all over thermodynamics is:

$$\frac{\partial x}{\partial y}\bigg|_z\frac{\partial y}{\partial z}\bigg|_x\frac{\partial z}{\partial x}\bigg|_y=-1$$

(where there exists $f(x,y,z)=0$)

The first approach of many people is to apply the chain rule, thereby not understanding where the minus sign came from.

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I think the problem is understanding the line between intuition and rigor.

The Riemann rearrangement theorem? Intuitive operation (you can just move around terms right?) unintuitive result.

Similar for things like $1+2+3+\cdots = -1/12$.

So with derivatives, it's like to think they are like fractions since $dx$ and $dy$ are like infinitescimals (and per Ben Crowell's answer even that viewpoint can be rigorized), but numerous theorems, such as the chain rule, are required to vindicate this view. Perhaps show a proof of the chain rule guided by our understanding of them as fractions, but remember to emphasize the technicalities it takes to underpin this.

Also note that the hypotheses are essential. These rules don't work when things aren't differentiable, for instance.

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I'd not want to tell students that there was a line between intuition and rigor, because I think that is a false distinction. It'd be bad to get people flinching every time they need to "move terms around", because mostly they should do so. One simply wants a refined intuition, not a belief that some sort of "technicalities" are waiting to ensnare us. The goal is not "avoiding trouble" but doing something constructive. – paul garrett Aug 28 '14 at 19:09
@paulgarrett so thinking of dy/dx as a fraction is unrefined intuition? – djechlin Aug 28 '14 at 19:22
I do think that it is a good, but unrefined intuition, yes. (Btw, for me, "fraction" and "ratio" and ... are approximate synonyms, despite arguable technical differences. I don't have space in my head to keep track of "technical details" :) ) But/and, then, the little troubles that a too-naive interpretation of "fraction" engender must be dealt with sanely, not over-reacting by throwing the baby out with the bathwater, and all that metaphor... From the other side, I don't think "rule-based" mathematics is very effective/potent, however "rigorous" it may be. Rules are fundamentally artificial. – paul garrett Aug 28 '14 at 22:40

I am not sure that it has the same rigorous foundation as Santiago Canez's answer, but I think that, if one wanted to argue for the mathematical legitimacy of the notation $\frac{\text dy}{\text dx}$, then one could view it (pointwise) as a quotient of elements of a tangent space $T$ that is 1-dimensional, for which therefore the scalar multiplication $\mathbb R \times (T \setminus \{0\}) \to T$ may be 'inverted' to give a division $T \times (T \setminus \{0\}) \to \mathbb R$. In this sense, it seems to me not so different from the common device of writing $Q - P$ for the vector $\vec{PQ}$ (i.e., viewing Euclidean space as an affine space under $\mathbb R^n$). Of course, such a viewpoint should never be explicitly mentioned in the calculus classroom; but I remember that it was actually advanced in a Monthly article that, unfortunately, I can no longer find.

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Although the above answers are good to great, I prefer a simpler answer.

If dy/dx is interpreted as a fraction, then it becomes 0/0, which is undefined and technically called an indeterminate form.

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text:
gamma {distributionsrd} R Documentation

## The Gamma distribution

### Description

Raw moments for the Gamma distribution.

### Usage

mgamma(
r = 0,
truncation = 0,
shape = 2,
rate = 1,
scale = 1/rate,
lower.tail = TRUE
)

### Arguments

 r rth raw moment of the distribution, defaults to 1. truncation lower truncation parameter, defaults to 0. shape, rate, scale shape, rate and scale of the distribution with default values of 2 and 1 respectively. lower.tail logical; if TRUE (default), moments are E[x^r|X ≤ y], otherwise, E[x^r|X > y]

### Details

Probability and Cumulative Distribution Function:

f(x) = \frac{1}{s^kΓ(k)}ω^{k-1}e^{-\frac{ω}{s}},\qquad F_X(x) = \frac{1}{Γ(k)}γ(k,\frac{ω}{s})

,

where Γ(x) stands for the upper incomplete gamma function function, while γ(s,x) stands for the lower incomplete Gamma function with upper bound x.

The y-bounded r-th raw moment of the distribution equals:

μ^r_y = \frac{s^{r}}{Γ(k)} Γ≤ft(r + k , \frac{y}{s} \right)

### Value

Provides the truncated rth raw moment of the distribution.

## The zeroth truncated moment is equivalent to the probability function pgamma(2,shape=2,rate=1) mgamma(truncation=2)

## The (truncated) first moment is equivalent to the mean of a (truncated) random sample, #for large enough samples. x = rgamma(1e5,shape=2,rate=1) mean(x) mgamma(r=1,lower.tail=FALSE)

sum(x[x>quantile(x,0.1)])/length(x) mgamma(r=1,truncation=quantile(x,0.1),lower.tail=FALSE)

[Package distributionsrd version 0.0.6 Index]


text:
p value from t value - online calculator

 Enter your $t$ value: Enter the degrees of freedom:
The test should be:


text:
# Adding contour lines to a 3D plot?

I am having trouble getting contour lines on a 3D plot (ListPlot3D). I have tried ListContourPlot3D, but that gives me a really weird looking graph with a ton of dfferent levels. What I would like is to get Contour lines on my ListPlot3D so it looks like a topo-map or something. As of now I have a contour map and a 3D plot - in essence I would like to combine them. Here is what the notebook currently looks like:

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Closely related question: can the color in MeshStyle be specified by a ColorFunction –  Jens Jan 26 '13 at 6:06

You are looking for MeshFunctions -> {#3 &}. The documentation for MeshFunctions lays out what values are accessible by a plot that uses MeshFunctions and its defaults in the Details section.

As to why ListContourPlot3D was not working for this: its purpose is entirely different. It is intended to draw contours for an $\mathbb{R}^3 \to \mathbb{R}$ function, such as $f(x, y, z) = x^2 + y^2 - z^2$.

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This is an unusual URL: wolfram.com/xid/0i1q1qqc4-mvz -- where did it come from? –  Mr.Wizard Jan 25 '13 at 21:32
@Mr.Wizard on the examples, click on the "In" and the URL is in the pop-up. Rm pointed out how to do it awhile back. –  rcollyer Jan 25 '13 at 23:54
@Mr.Wizard you get an empty pop-up where? on the website? –  rcollyer Jan 25 '13 at 23:56
It seems I misunderstood. I thought this was in the help Notebooks but I see it is on the Web copy. I knew that the code was there to copy but I've never noticed the URL at the bottom. Thanks! –  Mr.Wizard Jan 25 '13 at 23:58

Here is a method to obtain nice 2D contour plots using the Presentations Application (which I have a propriatary interest in). I don't have access to your data so I am using a regular function as an example.

<< Presentations

Draw3DItems[
{ContourDraw[Sin[x] + Sin[y], {x, -5, 5}, {y, -5, 5},
ColorFunction -> ColorData["StarryNightColors"]] //
RaiseTo3D[Sin[#1] + Sin[#2] &]},
NeutralLighting[0, .5, .1],
NiceRotation,
ImageSize -> 400]

The Presentations drawing commands allow us to deal directly with the graphics primitives and RaiseTo3D is a Presentations routine that does what it says. NaturalLighting is similar to the Mathematica Neutral Lighting, but gives much more control. NiceRotation gives mouse rotations without jumpings. Here is the graphic:

And in the notebook the contour lines even have Tooltips.

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How is this better than the default methods? Here you have to enter the function twice. Isn't it easier to just use Plot3D[Sin[x] + Sin[y], {x, -5, 5}, {y, -5, 5}, MeshFunctions -> {#3 &}, Mesh -> 7, ColorFunction -> ColorData["StarryNightColors"]]` ? –  Mr.Wizard Jan 28 '13 at 19:37
@Mr.Wizard There are two things that are better. This is a true contour plot with distinct colors between the contours. Also, using a true contour plot, it is possible to better specify the contour values and obtain distinct coloring in each region using the ContourColors function in Presentations. For example, one might want a number of closely valued contour lines in a plateau region. The Mesh method has mesh lines on top of a general shading. Secondly, the contours have tooltips, whereas the Mesh lines don't. I agree that for many cases the Mesh method might be sufficient. –  David Park Jan 28 '13 at 19:57


text:
# Transparent boundary conditions for evolution equations in infinite periodic strips

1 POEMS - Propagation des Ondes : Étude Mathématique et Simulation
Inria Saclay - Ile de France, UMA - Unité de Mathématiques Appliquées, CNRS - Centre National de la Recherche Scientifique : UMR7231
Abstract : We consider the solution of a generic equation $\gamma\rho(\mathbf{x})\partial^p_tu(\mathbf{x},t)-\Delta u(\mathbf{x},t) +V(\mathbf{x})u(\mathbf{x},t) = f(\mathbf{x},t)$, $\mathbf{x} = (x,y)$, for $t>0$, $p=1,2$ in a domain $\Omega$ which is infinite in $x$ and bounded in $y$. We assume that $f(\cdot,t)$ is supported for all $t>0$ in $\Omega_0 = \{\mathbf{x} \in \Omega \; | \; -a_- < x < a_+\}$ and that $\rho(\mathbf{x})$ and $V(\mathbf{x})$ are x-periodic in $\Omega \setminus \Omega_0$. We consider the associated $\theta$-scheme in time, to obtain a semidiscretized problem. We then show how to obtain for each time step exact boundary conditions on the vertical segments, $\Gamma_0^- = \{\mathbf{x}\in \Omega\; | \; x=-a_-\}$ and $\Gamma_0^+ = \{\mathbf{x}\in \Omega \;| \; x=a_+\}$, that will enable us to find the solution on $\Omega_0 \cup \Gamma_0^+ \cup \Gamma_0^-$. Then the solution can be extended in $\Omega$ in a straightforward manner from the values on $\Gamma_0^-$ and $\Gamma_0^+$. The method is based on the solution of local problems on a single periodicity cell, solved during an initialization step. The exact boundary conditions as well as the extension operators can be obtained for each time step through elementary computations using the solution of these local cell problems.
Document type :
Journal articles
Complete list of metadata

https://hal-ensta-paris.archives-ouvertes.fr//hal-00969313
Contributor : Aurélien Arnoux <>
Submitted on : Wednesday, April 2, 2014 - 1:16:39 PM
Last modification on : Thursday, January 14, 2021 - 11:56:04 AM

### Citation

Julien Coatléven. Transparent boundary conditions for evolution equations in infinite periodic strips. SIAM Journal on Scientific Computing, Society for Industrial and Applied Mathematics, 2012, 34 (3), pp.1563-1583. ⟨10.1137/110838030⟩. ⟨hal-00969313⟩

Record views


text:
Via Theorem 3.90:

\$$V=\\{1\\}\$$. \$$\mathrm{lim}\_\textbf{1}D=\\{(d_1)|d\_1\in D(1)\,\mathrm{and}\,D(\mathrm{id}\_1)(d_1)=d_1\\}\$$ together with a projection map \$$p_1((d_1))=d_1\$$. However, since \$$D\$$ is a functor, the constraint is always satisfied, and the projection map is the identity. So, the limit of \$$D:\textbf{1}\to\textbf{Set}\$$ is the set that it picks out in **Set**.

Via Definition 3.87:

Let \$$S\in\textbf{Set}\$$ be the set that \$$D\$$ picks out. A cone over \$$D\$$ consists of a set \$$C\$$ and a function \$$f:C\to S\$$ (the cone property is trivial here, just like the constraint in theorem 3.90). Morphisms between these cones are functions \$$a:C\to C'\$$ such that for \$$g:C'\to S\$$, \$$g\circ a=f\$$. A terminal object in this category **Cone**(\$$D\$$) is a set \$$T\$$ and a function \$$h:T\to S\$$ such that for every set \$$C\$$ and function \$$f:C\to S\$$ there is a unique \$$b:C\to T\$$ with \$$h\circ b=f\$$. For each element \$$s\in S\$$ there is a cone \$$\\{ * \\},f\_s:\\{ * \\}\to S\\}\$$ where \$$f_s(*)=s\$$. Each \$$b_s\$$ such that \$$h\circ b_s=f_s\$$ must pick out an element in \$$T\$$, since the images of each \$$f_s\$$ are distinct, the images of each \$$b_s\$$ must also be distinct, so \$$\lvert T\rvert\geq\lvert S\rvert\$$ and \$$h\$$ must be surjective. Uniqueness of \$$b_s\$$ forces \$$h\$$ to be injective, so \$$h\$$ is an isomorphism and \$$T\cong S\$$. This shows that following defintion 3.87 agrees with following theorem 3.90.


text:
# A 60 g piece of metal at an initial temperature of 92°C is transferred into 70 g of water initially at 24.5°C. The final temperature of the water and the metal is 40°C. How do you determine the specific heat of the metal?

## Assume perfect heat transfer.

Mar 16, 2017

$\frac{- 1.45362 J}{g r a m {C}^{o}}$

#### Explanation:

${q}_{\text{gain" = q_"lost}}$

(60g)(92^o - 40^oC)(x) = (70g)(24.5^oC - 40^oC)(4.18)"

4.18 is specific of water. Solve for x

$3120 x = - 4535.3$

$x = \frac{4535.3}{3120} \mathmr{and} \frac{45353}{31200}$

$x = - 1.45362$

This is the specific heat


text:
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Alberta: Athabasca AB, Chestermere AB, Raymond AB, Castor AB, Veteran AB, Sylvan Lake AB, AB Canada, T5K 4J6

Northwest Territories: Fort Good Hope NT, Yellowknife NT, Gameti NT, Salt Plains 195 NT, NT Canada, X1A 5L1

Saskatchewan: Plenty SK, Ituna SK, Cabri SK, Krydor SK, Lang SK, Alida SK, SK Canada, S4P 8C9

Manitoba: Souris MB, Manitou MB, Rapid City MB, MB Canada, R3B 5P8

Quebec: Saint-Cesaire QC, Abercorn QC, L'Assomption QC, L'Ile-Perrot QC, Longueuil QC, QC Canada, H2Y 6W5

New Brunswick: Centreville NB, Bathurst NB, Maisonnette NB, NB Canada, E3B 9H4

Nova Scotia: Annapolis NS, Inverness NS, Clare NS, NS Canada, B3J 7S7

Prince Edward Island: Bedeque and Area PE, Cardigan PE, St. Peters Bay PE, PE Canada, C1A 3N3

Newfoundland and Labrador: Arnold's Cove NL, Hare Bay NL, Fleur de Lys NL, Stephenville Crossing NL, NL Canada, A1B 8J4

Ontario: The Blue Mountains ON, Caledon ON, Norham ON, Riviera Estate, Limehouse ON, Maple Lane ON, Clear Creek ON, ON Canada, M7A 4L9

Nunavut: Ponds Inlet NU, Resolute NU, NU Canada, X0A 6H8

England: Corby ENG, Hastings ENG, Guildford ENG, Sutton Coldfield ENG, Altrincham ENG, ENG United Kingdom W1U 4A8

Northern Ireland: Bangor NIR, Newtownabbey NIR, Craigavon(incl. Lurgan, Portadown) NIR, Bangor NIR, Derry(Londonderry) NIR, NIR United Kingdom BT2 6H1

Scotland: Livingston SCO, Dunfermline SCO, Paisley SCO, Livingston SCO, East Kilbride SCO, SCO United Kingdom EH10 5B3

Wales: Cardiff WAL, Newport WAL, Cardiff WAL, Newport WAL, Newport WAL, WAL United Kingdom CF24 8D9


text:
# 1979 USAMO Problems/Problem 3

## Problem

$a_1, a_2, \ldots, a_n$ is an arbitrary sequence of positive integers. A member of the sequence is picked at random. Its value is $a$. Another member is picked at random, independently of the first. Its value is $b$. Then a third value, $c$. Show that the probability that $a + b +c$ is divisible by $3$ is at least $\frac14$.

## First Hint

The given problem is equivalent to proving that $4(x^3 + y^3 + z^3 + 6xyz) \ge (x + y + z)^3$.

## Second Hint

Try proving this inequality for all nonnegative real numbers x, y, z; not just positive integers.

## Final Hint

Prove that $x^3 + y^3 + z^3 + 3xyz \ge x^2(y+z) + y^2(x+z) + z^2(x+y).$

## Solution

Let x equal the probability of picking an element of the sequence equivalent to 0 mod 3, y equal those 1 mod 3, and z equal those 2 mod 3. Then, considering that 0+0+0, 1+1+1, 2+2+2, and 0+1+2 are divisible by 3, we obtain the equivalent inequality in the First Hint. This simplifies to $x^3 + y^3 + z^3 + 6xyz \ge x^2(y+z) + y^2(x+z) + z^2(x+y).$ Now, we attempt to prove that $x^3 + y^3 + z^3 + 3xyz \ge x^2(y+z) + y^2(x+z) + z^2(x+y).$ for all nonnegative real numbers x, y, z; our inequality will follow from this. But this is just Schur's Inequality with r = 1! This completes the proof.

Note: Schur's Inequality states that $x^r(x-y)(x-z) + y^r(y-z)(y-x) + z^r(z-x)(z-y) \ge 0$. This can be proven by setting WLOG $x \ge y \ge z$ and bunching together terms that are nonnegative. Challenge: when does equality hold? When does equality hold in the problem?