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# Pesticide pollution in freshwater paves the way for schistosomiasis transmission

## Abstract

Schistosomiasis is a severe neglected tropical disease caused by trematodes and transmitted by freshwater snails. Snails are known to be highly tolerant to agricultural pesticides. However, little attention has been paid to the ecological consequences of pesticide pollution in areas endemic for schistosomiasis, where people live in close contact with non-sanitized freshwaters. In complementary laboratory and field studies on Kenyan inland areas along Lake Victoria, we show that pesticide pollution is a major driver in increasing the occurrence of host snails and thus the risk of schistosomiasis transmission. In the laboratory, snails showed higher insecticide tolerance to commonly found pesticides than associated invertebrates, in particular to the neonicotinoid Imidacloprid and the organophosphate Diazinon. In the field, we demonstrated at 48 sites that snails were present exclusively in habitats characterized by pesticide pollution and eutrophication. Our analysis revealed that insensitive snails dominated over their less tolerant competitors. The study shows for the first time that in the field, pesticide concentrations considered “safe” in environmental risk assessment have indirect effects on human health. Thus we conclude there is a need for rethinking the environmental risk of low pesticide concentrations and of integrating agricultural mitigation measures in the control of schistosomiasis.

## Introduction

Schistosomiasis, also called bilharzia, is among the tropical diseases with the highest impact on socio-economic development, only exceeded by malaria. Approximately 218 million people are infected worldwide1. Infection has been strongly associated with long-term disabilities2. The number of deaths due to schistosomiasis is poorly documented with estimates ranging between 11,7003 to 280,000 each year4 because of hidden pathologies such as liver and kidney failure5. Schistosomiasis is caused by flatworms of the genus Schistosoma which parasitize humans as their definitive host (supporting the adult life stage of the parasite). The intermediate hosts are freshwater snails of the family planorbidae which release infective larval stages (cercariae) into the water. Transmission occurs when humans are exposed to water containing infected host snails; direct infection from person to person is not possible6. People are infected during routine agricultural, domestic, occupational and recreational activities, which expose them to infested water. Over 80% of afflicted people live in sub-Saharan Africa7, but the disease concerns public health in most (sub)tropical countries worldwide7 and has recently established in Europe8.

Control strategies against schistosomiasis focus on the treatment with praziquantel that kills the adult worms in the human host. However, even mass drug administration does not prevent re-infection in infested water, and schistosomiasis has been observed to rebound within short time9. For the sustainable control of schistosomiasis, it is also essential to interrupt the infection cycle by control of the intermediate hosts10,11,12. Host snails are susceptible to predation by organisms such as shrimps13 and ostracods14 which have been applied as biological control agents. Additionally, host snails are susceptible to competition from other snails and insects that feed on periphyton (microbes attached to surfaces), detritus and water plants15,16,17. Spreading of schistosomiasis has been often linked to the loss of biodiversity and the ecological degradation of freshwater habitats10,18,19. The findings suggest that host snails are significantly controlled by antagonistic species in natural habitats but that this ecosystem service is sensitive to anthropogenic impact. Therefore, it is essential to identify key environmental factors that drive the interactions of host snails with their associated community.

Recently, agricultural pesticides have returned to the focus of public attention as causes for the worldwide decline in insects and biodiversity20,21,22. Tropical regions, characterized by extensive agriculture and heavy rainfalls, are known areas of endemicity of schistosomiasis. In such conditions there is a high risk of surface run-off that washes pesticides from agricultural fields into adjacent freshwater systems23. However, information on pesticide concentrations in tropical freshwaters and their effects on the macroinvertebrate community are often fragmented and inadequate24,25. In temperate latitudes, snails are known as one of the macroinvertebrate taxa being most tolerant to pesticides26. In mesocosms, high concentrations of insecticides and herbicides favoured host snails indirectly through the reduction of predators and through the replacement of suspended algae with periphyton that serves as food for snails27. Additionally, even low levels of agricultural pesticides result in a typical replacement of sensitive macroinvertebrates by more tolerant taxa in mesocosms28 and in natural streams29,30. These effects are usually driven by insecticides that are most toxic to many macroinvertebrates30. Therefore, we hypothesized that pesticide pollution may favour highly tolerant snails that host human-pathogenic schistosomes over their more sensitive natural enemies and thus increase the risk of schistosomiasis transmission.

## Results

We studied how pesticide pollution and additional environmental factors affect the macroinvertebrate community composition in a typical endemic region of schistosomiasis. For this, we sampled 48 freshwater sites in the Kenyan Lake Victoria Basin (Fig. 1). The habitats ranged from small and medium-sized streams to irrigation channels, oxbow lakes, reservoirs and rice fields, and thus covered the main inland transmission sites in the study area31. Each site was monitored once during the rainy season in October 2017. To confirm the hypothesized high pesticide tolerance of host snails, we collected host snails and other common invertebrates and tested their acute sensitivity to two insecticides covering different modes of action.

### Pesticide tolerance of Schistosoma host snails

Among all macroinvertebrates tested, host snails of human-pathogenic schistosomes showed the highest tolerance to the neonicotinoid insecticide imidacloprid and to the organophosphorus insecticide diazinon (Fig. 2a,b), both of which were commonly found at the study sites. The acute median lethal concentration of imidacloprid after exposure for 24 h (LC5024h, concentration that killed 50% of test organisms) ranged from 0.007 mg/L for corixidae (true bugs) to 599 mg/L for the non-host snail Melanoides sp. The mortality of the host snails Bulinus africanus and Biomphalaria pfeifferi remained below 10% even at the highest concentration tested (165 mg/L) so that we could only estimate their respective LC50. This test concentration was close to the solubility limit of imidacloprid in water (610 mg/L32), indicating very high insecticide tolerance of the host snails. The LC5024h of diazinon for other taxa ranged from 0.5 µg/L for baetidae and caenidae (mayflies) to 13.5 mg/L for the non-host snail Ceratophallus sp. Again, the host snails B. africanus (15.2 mg/L) and B. pfeifferi (33.0 mg/L) were the most tolerant species.

### Pesticide pollution in the study area

The surveyed aquatic habitats showed considerable agricultural pesticide pollution. We analyzed 28 commonly applied active substances and degradation products and detected all the compounds in water samples, ranging from 5 to 27 (median = 20) substances per site. To quantify the toxicity, pesticide concentrations were converted to toxic units (TU) using the formula $$TU=lo{g}_{10}(\frac{Concentration}{LC{50}_{reference}})$$ with Concentration being the measured concentration of a pesticide and LC50reference being the LC50 of that pesticide for a standard reference organism (typically Daphnia magna, Table S1 in the Supplementary Material)33. Toxic units of −1 or −2 represent pesticide concentrations of 1/10 or 1/100 of the LC50, respectively. Toxic units of pesticides in agricultural European, Siberian and Australian streams typically reach from ≤−5 to −134,35.

In two sites all pesticides detected were below the limit of quantification such that the toxic unit of the most toxic compound (TUmax) could not be calculated. However, in the other sites, the TUmax ranged from −6.4 to −1.2 with a median TUmax of −2.3 (Table S2 in the Supplementary Material). The most toxic substances found were the organophosphorous and carbamate insecticides bendiocarb (most toxic substance at 17 sites, TU up to −1.66, Table S3 in the Supplementary Material), diazinon (most toxic at 15 sites, TU up to −1.71) and pirimiphos-methyl (most toxic at 5 sites, TU up to −1.21). Due to difficulties in the quantification of very low pesticide concentrations, the minimum TUmax was set to −5 for further analyses, a threshold at which typically no ecological effects have been observed in field studies29,34,35.

### Environmental factors driving host snail abundance

We investigated the influence of 27 environmental variables on the abundance of host snails, covering habitat type, land use, water chemistry and the composition of the macroinvertebrate community (Table S4 in the Supplementary Material). Host snails were found in 9 out of a total of 48 sites investigated in 2017; at one site they were infected with human-pathogenic schistosomes. The abundance of host snails encompasses the incidence, i.e. the probability of a population to occur at a given site, and the density of existing populations. Both endpoints can be driven by different environmental factors and thus were analyzed separately in a first step.

When each environmental variable was considered individually, the incidence of host snails increased significantly with pesticide toxicity (n = 48, χ² = 7.71, res. df = 46, p = 0.005, Fig. 3), species diversity (χ² = 4.42, res. df = 46, p = 0.035) and species richness (χ² = 4.39, res. df= 046, p = 0.036). Additionally, the incidence of host snails decreased with increasing dissolved oxygen (χ² = 8.06, res. df = 46, p = 0.004) and with the increasing dominance (proportion on all macroinvertebrates) of other grazers and herbivores that act as potential competitor species (χ² = 9.09, res. df = 46, p = 0.003; Table S4 in the Supplementary Material).

Environmental effects on the density of host snail populations were driven by a single stream (Table S5 in the Supplementary Material). This stream was characterized by extraordinarily high numbers of host snails and other macroinvertebrates. The site was located 100 m downstream of a bathing and washing area and was the only site at which infected host snails were found. When we excluded this site as an outlier, population density was explained only by a decrease in density with increasing turbidity (n = 8, χ² = 4.50, res. df = 6, p = 0.034).

In a second step, we combined the effects identified on the incidence and population density in a hurdle model in order to rank the relevance of the environmental variables in explaining the overall abundance of host snails. Stepwise regression identified that the incidence of host snails increased primarily with pesticide pollution, followed by an increase with the decreasing dominance of potential competitors, with increasing species richness and with a decreasing amount of dissolved oxygen (Table 1). The density of host snail populations only decreased with turbidity.

Results from stepwise regression are sensitive to the method used for model selection. Therefore, we additionally applied a multi-model approach by subjecting the full hurdle model to hierarchical partitioning (Fig. 4). Here the abundance of host snails was most strongly affected by the dominance of potential competitors, followed by the effect of pesticide pollution. In accordance with the results from backward selection, turbidity, species richness and dissolved oxygen showed intermediate effects, and species diversity was least important again.

In the next step, we confirmed the main drivers that potentially underlie the identified environmental variables using a principal component analysis (PCA). The first principal component explained 33.0% of the total variation among the sites and was associated with typical effects of surface run-off after heavy rainfall (Fig. 5): It increased with pesticide pollution and turbidity and with decreasing dominance of potential competitor species of the host snails. The second principal component additionally explained 29.4% of the variation and increased with species richness, with a decreasing amount of dissolved oxygen (indicating increasing oxygen consumption) and with decreasing dominance of potential competitors. Thus, the second principle component likely reflected an increase of host snails with eutrophication that supports more taxa but results in oxygen depletion. Moreover, the second principal component increased with the overall number of macroinvertebrate individuals as an indicator of productivity (n = 48, F = 8.12, res. df = 46, p = 0.007,  = 0.15) which further supported its interpretation as eutrophication. Host snails were only found when both the effects of run-off and eutrophication were high which resulted in a decreased dominance of potential competitors (Fig. 5).

### Ecological mechanisms

To better understand the ecological mechanisms through which pesticides affect host snails, we investigated effects of pesticides on the macroinvertebrate community composition. Pesticide pollution affected neither the dominance of all grazers (host snails and their potential competitors; n = 48, χ² = 0.41, res. df = 46, p = 0.520) nor of predators (χ² = 0.37, res. df = 46, p = 0.541) or other macroinvertebrates (χ² = 2.63, res. df = 46, p = 0.105). Thus, the overall distribution of grazers, predators and other taxa within the community did not significantly change with pesticide pollution (Fig. 6a). However, within the guild of grazers, pesticide pollution increased the dominance of snails (Fig. 6b) which were much more tolerant to pesticides than their highly sensitive insect competitors (Fig. 2). In contrast, pesticide pollution did not affect the composition of predatory macroinvertebrates (PERMANOVA; n = 48, F = 0.78, res. df = 46, p = 0.679) which generally showed intermediate sensitivity to pesticides (Fig. 2). Additionally, the taxonomic composition of potential predators had no effect on the balance of snails vs. potential competitors: The first principal component of a PCA on the composition of predators did not explain the dominance of snails within the grazers (n = 47, χ² = 0.02, res. df = 45, p = 0.890); the same was observed for higher principal components. Therefore, we conclude that pesticides indirectly favored host snails through negative effects on their competitors but not on their predators.

## Discussion

“To our knowledge this is the first field study providing evidence that ecological effects of agricultural pesticides can pose a serious risk to human health. Laboratory studies have already shown how pesticide pollution can increase risk of schistosomiasis27. We demonstrate a second mechanism with both field and laboratory data showing how such conditions favor the disease by benefitting the host snail. Host snails of schistosomiasis showed highest tolerance to insecticides amongst all tested macroinvertebrate taxa. Host snails were also solely found in habitats that were at least moderately affected by pesticides and eutrophication. In these conditions, snails replaced more sensitive potential competitor species. The study thus sheds light on an important risk factor for the transmission of schistosomiasis that has been largely overlooked in previous research on the ecology of host snails and in public health programs.

### Pesticide pollution in the study area

Concentrations of the highest exposure (TUmax) ranged from <−5 to −1.21, representing a 10th to <100,000th of the acute lethal median concentration for standard test organisms. This range of pesticide exposure is comparable to several previous studies conducted in agricultural streams of Europe and Australia22,30,35.

Given the predominance of subsistence farming in the study area, the results illustrate that pesticide pollution in freshwater is not limited to intensified agriculture. Apparently, the risk of pesticide runoff from agricultural fields is high in the study region. During the rainy seasons in April - June and in October - December, heavy rainfalls erode the cleared land, as indicated by the high turbidity observed in streams during sampling. Because the water bodies are typically not protected by riparian strips from surface run-off, pesticides are washed from flooded agricultural fields into the streams and reservoirs36. In addition, plant protection products are sold in Kenya at comparably low prices without the need for a certificate of competence. This makes pesticides available even for small farmers who may lack enough training and equipment to comply to the proposed environmentally safe use.

The study focused on a broad set of agricultural pesticides typically detected in water. It cannot be excluded that the overall pesticide toxicity might have been even higher due to additional compounds such as pyrethroid insecticides that require different analytical methods. However, pyrethroids typically occur concurrently with the compounds detected and show a similar range of toxicity35,37. In previous studies22,30,35 pesticides have been sampled during the peak exposure following run-off events after heavy rainfalls. Event-triggered sampling was not feasible in our study area. However, we sampled during the main rainy season at which we expected the highest pesticide exposure from run-off. Therefore, the TUmax values determined to characterize the toxic pressure in this study are comparable to those of previous investigations.

### Effects of pesticides on schistosomiasis infection

Host snails of human-pathogenic schistosomes were found exclusively in freshwaters that were at least moderately polluted with pesticides (TUmax >= −3) and at least mesotrophic. Physicochemical and land-use parameters had no significant effect on the abundance of host snails, and additional pollutants such as pharmaceuticals, personal care products and industrial chemicals have been shown to cause considerably lower environmental risk than pesticides at the study sites38. This was observed across various habitats ranging from reservoirs to irrigation channels and streams.

The results support our hypothesis that agricultural pesticide pollution in tropical freshwaters increases the risk of infection with schistosomiasis: Snails as intermediate hosts of human-pathogenic schistosomes are mandatory to close the infection cycle, and humans can become infected only from larval forms (cercariae) released by snails into the water6. Besides freshwater contamination with infected human excreta and human contact with freshwater infested with cercariae, presence of host snails is a major risk factor for transmission39. Two human-pathogenic trematodes occur in the study region: Schistosoma mansoni parasitizes snails of the genus Biomphalaria sp. and causes intestinal schistosomiasis, whereas S. haematobium parasitizes certain snails of the Bulinus africanus complex and causes urogenital schistosomiasis40. Access to sanitation is often insufficient in the densely populated study area11,41, and therefore many people are exposed to non-sanitized freshwater during activities such as bathing (particularly school children), washing and field work39. This is especially pronounced at the shore of Lake Victoria which suffers from a high disease burden11,42. When traveling, schistosomiasis transmission may be imported to inland areas if host snails are present43. In these conditions, we expect that the risk of infection is influenced by the occurrence of intermediate host snails.

Our finding of pesticide-induced shifts in the community composition towards more snails are in line with various studies that reported significant ecological effects of pesticides even in streams with low concentrations resembling a TUmax of −2 to −4. These effects include changes in the macroinvertebrate community composition towards more tolerant taxa30, reduced leaf litter breakdown37 and the development of pesticide resistance35. The results, however, contrast the common perception of the environmental risk of pesticides. For example, according to the European framework for the registration of plant protection products, environmental concentrations <1% of the acute LC50 of the most sensitive standard reference organism are generally considered safe44; this would resemble a TUmax up to −2. No such threshold concentrations have been defined in Kenya, but pesticides need to be considered environmentally safe by the national Pest Control Product Board before registration45. The present study shows that pesticides nevertheless affect the community composition of freshwater macroinvertebrates and that these ecological effects can have serious consequences for human health, hence the need for revision of acceptable regulatory concentrations.

### Ecological mechanisms supporting host snails

Given the very high pesticide tolerance of snails compared to the pesticide concentrations measured in the environment, a direct effect of pesticides on the observed host snails appears unlikely. Instead, pesticides may indirectly favor host snails through adverse effects on their antagonistic species such as predators and competitors. A recent mesocosm study showed that herbicides and insecticides can favor host snails of human-pathogenic schistosomes through effects on predators and the support of periphyton as food source for snails (through effects on antagonistic planktonic algae)27. However, the study did not collect field data, nor did it discuss potential effects on competitor species27. The observed effects on predators in the mesocosm study contrast our results, probably because pesticide concentrations were 3–4 orders of magnitude higher than those observed at our study sites. Such concentrations may affect even taxa such as predators that showed generally intermediate pesticide tolerance in our tests and in previous studies46. We observed that with increasing pesticide pollution snails replaced grazing insects that are known to compete with the more tolerant snails16 and are generally highly sensitive to insecticides and some fungicides46,47. In fact, pesticides have been shown to affect the survival and emergence of aquatic insects at concentrations down to 0.005% (4 orders of magnitude below) of their acute LC5048,49. Moreover, the calculation of toxic units for additional trophic levels revealed highest risk for insects and crustaceans compared to algae and vertebrates in our study sites38. Therefore, we focused on the toxicity of pesticides to invertebrates. Our results indicate that in the field, pesticides favor snails mainly through negative effects on more sensitive competitors. Pesticide pollution was closely related with turbidity, as both factors increase with rainfall induced flooding23,50. Nevertheless, these factors showed contrasting effects on the abundance of host snails (increased incidence vs. decreased population density). We hypothesize that flooding results in a short-term reduction of host snail populations due to increased flow velocity51, whereas pesticide exposure in the long-term facilitates the establishment of tolerant taxa such as snails30,52. This may explain the generally low population densities of host snails observed during the rainy season and the more obvious link of pesticide pollution with incidence than with population density.

## Conclusions

The present case study illustrates that serious consequences of agricultural pesticide pollution arises for public health, even at concentrations considered safe within the traditional risk assessment. Given that pesticide application – particularly in developing countries - is predicted to increase 2- to 5-fold from 2000 to 2050 to meet the food demand of a growing human population, freshwater pollution and its ecological effects will aggravate53,54. The results underline the urgent need for reassessing the environmental risk of low pesticide concentrations and for integrated disease management that includes a focus on the regulation and management of pesticides in areas where schistosomiasis is endemic or might be introduced due to potentially favorable ecological conditions.

## Materials and Methods

### Acute toxicity tests

We investigated acute insecticide sensitivity for all macroinvertebrate taxa that could be found in sufficient quantities in October/November 2018 from 6 sites in the study area. Site selection was based on host snail availability, high macroinvertebrate diversity, and low presumed pesticide pollution as indicated by buffer strips to minimize testing populations that may have developed pesticide resistance35. Test organisms were collected using sweep nets, standard pint dippers and snail catchers; they were sorted and identified to family level in the field. The organisms were placed in plastic lunch boxes filled with water from the sampling site and aerated using battery-operated air pumps. The containers were cooled in a portable fridge at 18 °C in order to prevent mortality during transport to the laboratory. The organisms were acclimatized to test conditions overnight. Tests were performed in a shaded screen-house with temperatures ranging from 20 to 33 °C. Commonly applied agricultural insecticides comprise three major classes with distinct modes of action: Organophosphates/carbamates, neonicotinoids and pyrethroids. If sufficient test organisms were available, we tested the acute sensitivity to one of the most toxic substances among the organophosphates (diazinon) and neonicotinoids (imidacloprid) measured at the study sites, respectively. To increase environmental realism, we applied local plant protection products containing the active substance and additional carriers that might have affected the toxicity.

Tests were performed according to the Rapid Test protocol for field-collected organisms55 with minor modifications. Imidacloprid was applied as a 70% wettable granule formulation (Loyalty, manufactured by Shandong United Pesticide Industry Co., Ltd. Jinan city, China) and diazinon was applied as a 60% emulsified liquid formulation (Diazol, repacked and distributed by Laibuta Chemicals Ltd). Fresh stock solutions were prepared a few hours before each test dissolving the formulations in a 1:1 mixture of bottled water and activated carbon filtered stream water. This mixture was a compromise to minimize adverse effects from water to which the organisms had not been adapted, and to minimize potential effects from residual toxicity and from dissolved solids in the stream water which can absorb pesticides. Stock solutions of 165. mg active substance/L were left to stir overnight in amber vials; no additional solvents were applied.

Test organisms were exposed to 6 test pesticide concentrations including a control. For each taxon test concentrations were selected from the following geometric series such that they covered the expected range of a partial response from <5% mortality in the lowest concentrations to >95% mortality in the highest concentration: 0.001; 0.004; 0.014; 0.055; 0.209; 0.792; 3.01; 11.4; 43.5, 165 mg/L.The ranges expected to result in a partial response were identified from data bases32,56 and previous studies35 for related taxa and substances. The tests were performed in 100 ml glass vessels containing up to 5 individuals of the same species (predators were kept individually to avoid cannibalism). The test medium was constantly aerated using aquarium pumps connected to glass pipettes via a silicone tube. Only the glass pipette had contact with the test solution and the air flow was controlled through a clamp on the silicone tube. After 24 h and = 48 h, mobile, immobilized and dead individuals were counted. Individuals were considered immobilized when no movement was observed within 10 s of undisturbed observation or after probing with a rod; fanning of gills was not considered movement.

### Field sampling

48 study sites located in Homa Bay, Kericho, Kisii, Kisumu, Migori and Nyamira in Western Kenya were investigated from September–November 2017. The aquatic habitats were chosen from areas characterised by different types of land use and crops grown, identified using aerial photos from Google Maps. We classified the sampling sites according to habitat types (major tributary, minor tributary, irrigation channel, oxbow lake, reservoir or rice field) and the surrounding dominant land use within 50–100 m (natural, agricultural, semi-urban, urban or industrial). Agricultural land use was classified by farm type, subsistence or irrigation schemes and crop type (maize, tea, sugarcane or rice).

Streams, rivers and oxbow lakes were sampled across a 50-metre section whereas dams and irrigation schemes were sampled at four sub-sites. The aquatic habitats were surveyed for the presence of submerged, emerging or floating vegetation as well as algal bloom and the percentage of detritus cover. Depth was measured at the bank at point of sampling using a metre rule or was indicated as >1 m. Flow velocity was estimated with the drift approach. Additionally, we measured physicochemical parameters (temperature, conductivity, pH, dissolved oxygen, carbonate hardness, ammonium, phosphate, nitrate, nitrite and nitrite) and the turbidity on site using cholorimetric test kits (MACHEREY-NAGEL Quantofix, Düren, Germany), a turbidimeter (WTW TURB 355 IR, Weilheim, Germany), a multi-measurement probe (EXTECH ExStick EC500, Boston, USA) and an oxygen probe (EXTECH ExStick DO600, Boston, USA).

For pesticide analyses, grab samples were taken using pre-cleaned glass beakers. Briefly, oven dried 500 mL beakers were rinsed three times with the sample water and filled up to the top. After suspended solids settled, 1 mL aliquots were taken into five 2 mL autosampler amber glass vials (Phenomenex, Germany) using a volumetric pipette. All samples were immediately stored in a portable freezer (Waeco Compressor Cooler Box – 50 litres #CF-50) at −4 °C and transferred to the laboratory where they were stored at −20 °C until analysis. For quality control, sampling and trip blanks were taken during each sampling campaign.

Macroinvertebrates were sampled along four equal sections of the water body. Banks were sampled in a criss-cross fashion along the sampling points using littoral sweep nets, dippers and snail catchers; a standardised sampling procedure was predetermined to collect macroinvertebrates comprehensively within the different microhabitats and habitat types. In brief, each site was sampled for 30 minutes by two persons in parallel. Collected macroinvertebrates were sorted and counted in white plastic trays and preserved in 70% ethanol. Some host snails were transported to the laboratory and checked for Schistosoma infection. The snails were kept individually in a 24-well plate (Nunc 142475 Nunclon) and exposed to sunlight for a minimum of 30 minutes, and shed cercariae were = identified under a dissecting microscope (Zeiss AxioCam5 100–400x) and an identification key for cercariae57. Macroinvertebrates were identified under a dissecting microscope (Zeiss AxioCam5 100–400x) to the lowest taxonomic level possible with the available identification keys58,59,60,61,62,63,64,65,66,67. Based on these data we calculated the following biological indices: the overall macroinvertebrate individual number, the species richness, Pielou’s species evenness, the Shannon index for species diversity, the ASPT indicator for stream health from the South African Scoring System SASS68, and the dominance (relative abundance) of potential predator and competitor species of the host snails. We considered all taxa as potential predators that comprise more than a marginal proportion of predatory species in the study region that may feed on freshwater snails or their eggs. Similarly, we considered all taxa as potential competitors that comprise more than a marginal proportion of periphyton feeders or herbivores in the study area (Table S6).

### Pesticide analysis

Details on the analysis of pesticide residues in water samples have been described in Kandie et al.38. In brief, 25 μL of an internal standard solution containing 40 isotope-labelled compounds (40 ng/mL9, 25 µL of methanol and 10 μL of 2MNH4-formate buffer (pH 3.5) was added to each sample prior to instrumental analysis. Analysis was performed using high-performance liquid chromatography (HPLC, Ultimate 3000 LC) coupled to high resolution mass spectrometry (HRMS, QExactive Plus MS) from Thermo Scientific. The sample (100 µL) was directly injected for chromatographic separation (Phenomenex Kinetex c18 EVO, 50 × 2.1 mm, 2.6 µm particle size), equipped with a pre-column (5.0 × 2.1 mm) and 0.2 µm in-line filter using a methanol/water gradient containing 0.1% formic acid. Heated electrospray ionisation (ESI) was performed for both the negative and positive modes with combined full scan run (100–1500 m/z) at a nominal resolving power of 70,000 (referenced to m/z 200) and data-independent MS/MS fragmentation (DIA) at a nominal resolving power of 35,000. An isolation mass window of m/z = 50 (m/z range 122–860) or m/z = 260 (m/z range 860–1370) was used in DIA analysis. Matrix matched calibration standards were prepared for 11 calibration levels (ranging from 1 to 2,000 ng/L) using 1 mL filtered water from a pristine reference stream (Wormsgraben, Harz Mountains, Germany). Quantification of detected pesticides was performed using isotope–labeled internal standards of compounds with closest retention time to the target compound. Data evaluation was performed using MZmine (Version 2.38 http://mzmine.github.io/) and trace finder (Thermo, version 4.1 https://www.thermofisher.com/ke/en/home/industrial/mass-spectrometry/liquid-chromatography-mass-spectrometry-lc-ms/lc-ms-software/lc-ms-data-acquisition-software/tracefinder-software.html).

### Data analysis

All data were analyzed using the software R 3.5.269. From the mortality observed in the acute toxicity tests we calculated the acute lethal median concentrations after exposure for 24 h (LC5024h) with 4-parameter log-logistic non-linear regression available with the package drc 3.0–170. The parameters for the upper and lower boundary were fixed to 1 and 0, respectively. If a taxon had been tested at more than one date or from more than one site, data were merged prior to the analysis. Tests which showed >30% control mortality were excluded from analyses. The resulting LC50values were ranked in ascending order to obtain the species sensitivity distribution (SSD). This increase in the proportion of affected taxa with pesticide concentration was described using a quasibinomial generalized linear model (GLM) with a logit link function.

For all GLMs in this publication, p-values were obtained from likelihood-ratio tests that compared each model to a null model without the environmental variable. Depicted means and 95% confidence intervals were extracted from (generalized) linear models using the package effects 4.1-071. Normally distributed residuals and homoscedasticity were confirmed using normal Q-Q plots and plotting residuals vs. fitted values; GLMs were inspected using scaled residuals available with the package DHARMa 0.2.072. The effects of each environmental variable measured on the incidence of host snails were analyzed using one-way binomial GLMs (binary regression) with a complementary log-log link function which allows for a non-symmetric dose-response curve. The effects of each environmental variable measured on the density of existing host snail populations were analyzed using one-way GLMs with a zero-truncated negative binomial distribution of residuals and a log link function available with the package VGAM 1.0-673. This way we dealt with overdispersion and with the missing possibility for the population density to be zero. Because many effects on the population density of host snails were driven by a single site (site 39) with extraordinarily high numbers of host snails and other macroinvertebrates, we repeated the analysis on the effects of host snail density with that site excluded. Only those environmental variables were considered in further analyses which significantly (p < 0.05) explained the population density after site 39 had been excluded.

Environmental variables that explained the incidence or population density of host snails were combined in a hurdle model available with the package pscl 1.5.274. Hurdle models consist of two connected generalized linear models to simultaneously fit the incidence (zero part) and the population density (count part). Prior to modeling, the environmental variables were standardized (normalized and centered) to make the model parameters comparable. Environmental variables that significantly (p < 0.05) explained the incidence were incorporated in the zero part of the model (binomial GLM with complementary loglog link), and environmental variables that explained the population density were incorporated in the count part (zero-truncated negative binomial GLM with log link). To avoid overfitting, we applied an additive model without interactions. Then we sequentially removed all non-significant environmental variables based on a likelihood ratio test (backward elimination). Each time a variable had been removed, we started testing again with the least-significant of the remaining variables according to the model statistics.

Additionally, we fitted a hurdle model with all the environmental variables that on their own significantly explained the incidence or the population density included in both the zero and the count part. Hurdle models consist of two connected generalized linear models to simultaneously fit the incidence (zero part) and the population density (count part)74. We removed all non-significant effects from this full model in a stepwise backward-elimination process and then sorted the remaining effects based on the magnitude of their regression coefficients. Due to multicollinearity, selecting a single minimum adequate model can lead to different results depending on the method of model selection75. Therefore, this model was subjected to hierarchical partitioning. Because this procedure is currently not available for hurdle models in R, we extended the code of the function hier.part from the package hier.part 1.0–476. The modified function started with a null model and each step included an environmental variable to both the zero and the count part at the same time. The improvement of the goodness-of-fit that resulted from the inclusion of an environmental variable was quantified using the log-likelihood.

Relations among the environmental explanatory variables were visualized using a principal component analysis (PCA) available with the function prcomp in basic R. A PCA reduces complexity by combining correlated environmental variables to few “supervariables” called principal components. The data were standardized prior to the analysis. Additionally, the association of the second principal component with the log-transformed number of macroinvertebrate individuals was analyzed using ordinary one-way linear regression.

The effect of pesticide pollution (TUmax) on the overall community composition consisting of grazers, potential predators and other macroinvertebrates was analyzed using a permutational multivariate analysis of variance (PERMANOVA) available with the package vegan 2.5–477. We also investigated the effect of pesticide pollution on the taxonomic composition of potential predators using a PERMANOVA. To investigate effects of the species composition of potential predators on the grazer composition, we performed a PCA on the predator composition. The first to fifth principal component was then fitted vs. the dominance of snails within grazers using quasi-binomial GLMs with a logit link function to account for the possibility of overdispersion. The proportions were weighted with the numbers of observed grazers. Similarly, the effects of pesticide pollution on the dominance of all grazers, of potential predators, and of snails within the guild of grazers were analyzed using quasi-binomial GLMs with a logit link function. The proportions were weighted with the numbers of observed individuals or of observed grazers, respectively. All data were analyzed using the software R 3.5.269.

## Change history

• ### 07 April 2020

An amendment to this paper has been published and can be accessed via a link at the top of the paper.

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## Acknowledgements

This work was supported by the DFG (Deutsche Forschungsgemeinschaft), grant number LI 1708/4-1, BR 2931/3-1, HO 3330/12-1 and the German Helmholtz long-range strategic research funding. We also gratefully acknowledge the financial support for this research by icipe’s core donors, the UK’s Department for International Development (DFID); Swedish International Development Cooperation Agency (Sida); the Swiss Agency for Development and Cooperation (SDC); Federal Democratic Republic of Ethiopia; and the Kenyan Government. We thank Dr Saskia Knillmann for her support in the preparation of this research project. We would also like to thank the various free software available scientist that were used such as the GIS software DIVA-GIS and the statistical program R.

## Author information

Authors

### Contributions

M.L. conceived the approach; M.L., J.B., U.F., W.B., B.T., E.A., H.H., F.M., A.A. and F.K. designed the research; A.G., F.K., L.M., J.B. and J.A. conducted the research; J.B. and A.G., analysed and interpreted the data; J.B. and A.G., drafted the initial version; J.B., A.G., F.K., M.L., U.F., W.B., B.T., E.A. and H.H. contributed to the final version - all approved the final version of the publication.

### Corresponding authors

Correspondence to Akbar A. Ganatra or Ulrike Fillinger or Matthias Liess.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

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

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Reprints and Permissions

Becker, J.M., Ganatra, A.A., Kandie, F. et al. Pesticide pollution in freshwater paves the way for schistosomiasis transmission. Sci Rep 10, 3650 (2020). https://doi.org/10.1038/s41598-020-60654-7


text:
# Talk:Sign function

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## removal announcement

I removed the following

Also, if the step function h0(x) is thought of as a mathematical switch, with h0(x) = 0, then the signum function can be expressed as
$\sgn x = h_0 (|x|) (-1)^{h_0(-x)}.$

as I do not understand what it adds.--Henrygb 14:34, 29 Mar 2005 (UTC)

Since h(x) can be different things, it looks like someone was just leaving a note of how to calculate sgn for people who are used to using another definition of h(x)? Who added it in the first place? - Omegatron 16:12, Mar 29, 2005 (UTC)

isn't it kinda dumb to have this and the signum function page, which are talking about the same thing????Scythe33

In the first paragraph, the word "sign" in "extracts the sign of a real number" was linked to the Negative and non-negative numbers page. There doesn't seem to be a page for the correct concept. I have removed the link. Iggle 08:27, 27 March 2006 (UTC)

## Pronunciation?

Does anyone know the correct pronunciation of "signum"? I'd intuitively say "sigg-numb", but in context, "sigh-numb" might also be correct. Might be worth putting in.

Hmmmmm... I checked Wiktionary, but it didn't provide a pronunciation. And dictionary.com provided no entries. I usually pronounce it "sigg-numb," like you. -- He Who Is[ Talk ] 18:17, 22 June 2006 (UTC)

Since it is (presumably) Latin, the "g" should be pronounced. Zaslav (talk) 01:09, 27 February 2008 (UTC)
"Signum" pronounces as "SIN-yoom". How to Pronounce Latin --Octra Bond (talk) 05:48, 8 June 2008 (UTC)
The pronunciation given at that site is the pronunciation of Ecclesiastical Latin, as is made quite clear there. If the word signum is pronounced as part of Latin liturgy, for instance when reading in the liturgy of Christmas from the Gospel of Luke 2:12 Et hoc vobis signum : invenietis infantem pannis involutum, et positum in præsepio, you can expect to hear "SEEN-yoom" (or, if the speaker is Italian, "SEEN-yoom-uh"). In Classical Latin, as far as we know, the 'g' was hard, as in "SEEGG-noom". In addressing an English-speaking audience, I'd go with "SIGG-numb", as if the word has become an accepted English word, in analogy to how the 'g' is pronounced in 'magnificent' and 'recognition'.  --Lambiam 19:22, 8 June 2008 (UTC)

## Complex Signum

I've added a section on the signum of complex numbers. Feel free to critique or remove, if you feel it is extrenuous. -- He Who Is[ Talk ] 18:14, 22 June 2006 (UTC)

• I've just added a description of csgn from Maple, which is another generalization of signum to complex numbers. I think it was neccessary because csgn redirects here but there is no mention of csgn. Rjgodoy 02:44, 30 May 2007 (UTC)

## Signature Function (permutations)?

Sgn redirects here, but there's no mention of the signature of a permutation matrix which is also denoted with the sgn() function. That's actually what I was trying to find, and only happened to stumble across it linked from another article. 142.59.195.50 19:36, 31 March 2007 (UTC)

• Hello, and thanks for the comment. I have added a note about sgn(σ) that suggest Levi-Civita symbol. The generalization to n dimensions of Levi-Civita pseudotensor is another form for the sgn(σ) function you refer. Rjgodoy 02:29, 30 May 2007 (UTC)

## Generalized signum function

The first paragraph of this section says "$~\varepsilon(x)^2 =1~$ everywhere, including at the point $~x=0~$" but the second says "$~\varepsilon(x)~$ cannot be evaluated at $~x=0~$". Why does it conflict itself? --Octra Bond (talk) 05:55, 8 June 2008 (UTC)

Generalized functions do not behave like ordinary functions. The paragraph pertains to a construction in the algebra of generalized functions developed by Yu. M. Shirokov; see Wikisource:Algebra of generalized functions (Shirokov). In that algebra, δ(x)2 = 0 at x = 0 even though δ(0) = +infinity. The notation is (in my opinion) misleading, since what is being multiplied are the generalized functions themselves, and a notationally clearer statement would have been that ε)(x) = 1 and δ)(x) = 0 for all x, in which denotes the (non-commutative) operation of multiplication for generalized functions.  --Lambiam 18:40, 8 June 2008 (UTC)

## Undefined?

The textbook "Calculus- a complete course" by Robert A. Adams says that sgn(x) is equal to 1 if x > 0; equal to -1 if x < 0 and that it is undefined when x is equal to 0; that is to say that:

$\sgn x = \begin{cases} -1 & \text{if } x < 0, \\ undefined & \text{if } x = 0, \\ 1 & \text{if } x > 0. \end{cases}$

The explanation is verbatim "The value of the sgn(x) tells whether x is positive or negative. Since 0 is neither positive nor negative, sgn(0) is not defined." This makes sense seeing as how the sign function is x/(abs(x)), and when x = 0 one would get 0/0. This directly contradicts this article, which both has a dot at (0,0) and states that:

$\sgn x = \begin{cases} -1 & \text{if } x < 0, \\ 0 & \text{if } x = 0, \\ 1 & \text{if } x > 0. \end{cases}$

Someone mentioned above that since the signum function is a generalized function, sgn(0)=0 can apply but that doesn't mean that it is true. If somebody knows anything about this, I'd like this to be disambiguated and (if wrong) fixed. A small explanation about 'why' it isn't undefined would also be nice if somebody understands that. --BiT (talk) 14:25, 10 November 2008 (UTC)

This, this, this and this page from Matlab are sources that sgn(0) is defined, but isn't a published textbook more reliable than some articles on the internet? --BiT (talk) 14:32, 10 November 2008 (UTC)

Leaving it undefined at 0 makes the derivative make more sense (as it too would be undefined at x=0). —Preceding unsigned comment added by 142.162.24.209 (talk) 04:59, 5 February 2009 (UTC)

Leaving it undefined at 0 makes no difference for the derivative since limits are defined to require an approach from both sides. Defining the sgn to be 0 at 0 allows for a more fluid definition of the matrix determinant in terms of the Leibniz formula. Doing so allows the sum to be taken over all arrangements up to N, and not just all permutations. It makes more sense from a combinatorial point of view. Antares5245 (talk) 00:20, 14 December 2009 (UTC)

Ancient post, but here goes, first the complicated answer: The two functions are equivalent under the L2 norm since they differ in only one point. This implies for example that the fourier series for both is the same. One possible reason to favor the sgn(0)=0 convention is that the fourier series for sgn(x) converges (pointwise) to 0 in x=0. Making a distinction though between the two functions is mostly irrelevant. Almost everyone uses the zero convention or states explicitly what the behaviour at zero is.--217.84.27.165 (talk) 11:13, 13 November 2011 (UTC)

## math on Wikipedia

Is it just me or is math on Wikipedia either exceedingly complicated it's not worth trying to understand, or such within the realm of basic common sense it is laughable? "Any real number can be expressed as the product of its absolute value and its sign function"

173.183.79.81 (talk) 23:45, 15 March 2011 (UTC)

It's not just you and not just Wikipedia; all mathematics falls into one of these two categories. --catslash (talk) 22:12, 18 March 2011 (UTC)
173, sign in and help make it better. Cliff (talk) 05:20, 1 June 2011 (UTC)
It may be common sense, but it is well worth stating basic mathematical axioms in an encyclopedia. You may as well say "I could have thought of the whole concept of signum, and therefore it doesn't need an article." Indeed, signum is very very simple, and anyone with common sense could have come up with its definition, drawn the simple graph on the article, worked out the axioms and its relationship to the absolute value function, etc. But it is still useful to have such simple definitions available, so we can derive more complex things from then. The axiom you quote is used in this article to show a few other interesting properties. —MattGiuca (talk) 07:04, 25 August 2011 (UTC)

## Algebraic representation

As far as I can tell, the entire point of the section on algebraic representation is "couldn't we express the signum function without using a conditional?" In other words, the basic definition of signum has a three-way switch using the condition brace; can we do it using normal (non-conditional) mathematics?

It seems like it could be simplified as $\frac{x}{|x|}$ or $\frac{x}{\sqrt{x^2}}$, but I suppose that would be undefined when x is 0. (This would lend credence to the above suggestion that signum be undefined when x is 0.)

Failing that, I don't really see the point of this section. It doesn't cite any sources to show where this complicated formula came from. The fact that it requires n be a certain number of decimal digits is mathematically messy and requires more special cases than the simple definition of signum in the first place. What is the point / use of this formula? —MattGiuca (talk) 07:17, 25 August 2011 (UTC)

I thought just the same. Your comment got no response in 3 months, so I deleted the section. Stephanwehner (talk) 07:01, 3 December 2011 (UTC)

## Citation needed?

Really? Is a citation really needed for the derivative? Surely this is a basic calculation. It follows trivially from the line below it about the Heaviside step function(which doesn't say citation needed).--217.84.27.165 (talk) 11:21, 13 November 2011 (UTC)

The same thought occurred to me - that this is essentially the definition of the delta function. However, it could be that mathematicians have scruples about differentiating a discontinuous function, which I lack. Perhaps KlappCK [1] could explain the problem? --catslash (talk) 00:05, 14 November 2011 (UTC)
The purpose of a citation here is to increase the reliability of the article and to solidify it's relation to other functions. I have had to derive the result myself, starting with the Heaviside step function, so I am comfortable with the definition, but this page isn't just for me, or either of you, for that matter. Furthermore, the derivative is not universally defined (if you have ever used Mathematica, D[Sign[x],x] returns Sign'[x] because it does not "know" the derivative of the sign function). However, one could point to the definition for the sign function in terms of the Heaviside step function [2] and then to the definition of the Heaviside step function's derivative [3] in a footnote showing how these two pieces of information and simple differentiation yield the result with a link to said footnote next to definition in the body of the article. This would seem satisfactory to me, anyway. KlappCK (talk) 20:09, 27 December 2011 (UTC)
As an aside here, I believe that it may be beneficial for overall readability to move the definition of the sign function in terms of the step function up about the definition of the derivative of the sign function and use the former definition to derive the latter.KlappCK (talk) 20:29, 27 December 2011 (UTC)
I made the changes I had proposed. See what you think.KlappCK (talk) 21:25, 27 December 2011 (UTC)

## Article title

Isn't "signum function" the original and still most common name for this function? I think the lead section of the article should be changed to something like: "In mathematics, the signum function (from signum, Latin for "sign") is an odd mathematical function that extracts the sign of a real number. This function is also sometimes called the sign function, although this may lead to confusion with the sine function. In mathematical expressions, the signum function is often represented as sgn."

Also, I think the article title should be "Signum function", not "Sign function". Isheden (talk) 16:59, 7 March 2012 (UTC)

format: plain


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# Outside/Inside View

The terminology of outside/inside view was coined by Daniel Kahneman and Dan Lovallo in 1993 (Kahneman and Lovallo, n.d.), whose main claim is this:

Decision makers are excessively prone to treat problems as unique, neglecting both the statistics of the past and the multiple opportunities in the future.

These are the result of two biases.

1. The forecasts of future outcomes are anchored on plans and scenarios of success, rather than on past results, and are hence overly optimistic
2. Evaluations of single risky prospects neglect the possibility of pooling risks

There are two distinct modes applied to forecasting. The inside view of the problem focuses on the case at hand, by considering the plans and obstacles to its completion, and constructing scenarios of future progress. This is the mode commonly adopted.

The outside view ignores the details of the case at hand, and focuses on statistics of cases similar to the present one.

When making a forecast, it is useful to consider the outside view first to obtain a ball-park number. Then, use the insider view to adjust the previous estimation up and down.

## Bibliography

Kahneman, Daniel, and Dan Lovallo. n.d. “Timid Choices and Bold Forecasts: A Cognitive Perspective on Risk Taking.” INFORMS. https://doi.org/10.1287/mnsc.39.1.17.


text:
# How much energy does it take to flip a spin in a ferromagnet?

Sorry the question is vague, I don't know what the name of the theory is that describes this so I don't know what to google. No need to explain anything just link me the wiki page pls.

The energy to flip a single spin is the interatomic exchange interaction $J$ times the number of magnetic neighbors. If you want a wikipedia article, this might be useful: https://en.wikipedia.org/wiki/Ising_model

Or this one: https://en.wikipedia.org/wiki/Superexchange

The best way to measure these energies directly used to be inelastic neutron diffraction. Now there is also inelastic x-ray scattering.


text:
# Categorical Variables in Trees I

I find it remarkable that very few of the current implementations of tree algorithms in R exploit an important “trick” which was mentioned in the original seminal paper in 1984!

“For a two-class problem, the search for the best categorical split can be reduced to M steps using the Gini criterion”

Breiman, L., Friedman, J., Olshen, R. and Stone, C. (1984). Classification and Regression Trees , Wadsworth International Group

This implies that instead of having to search $$2^k-1$$ combinations of the $$k$$ levels it is sufficient to try just k of them !

The practical consequences of this exponential vs. linear scaling are quite grave, especially with modern datasets that often contain lots of categorical variables with many (not rare to see $$k> 1000$$) levels.

Of the popular packages that I tried, only rpart, gbm and RWeka avoid the “exponential trap” and pose (almost) no limit on the number of levels:

#read in the Rossman data set from this kaggle competition:
#  https://www.kaggle.com/c/rossmann-store-sales
#train = read.csv('H:/kaggle/Rossmann/input/train.csv.gz',as.is=T)

#  oh well, fake data serve to illustrate the point just as well:
N=5*10^5
train = cbind.data.frame(Sales= 0, Store=sample(1:1000,N,replace=TRUE))
train$Sales = 0.1*train$Store +rnorm(N)
train$Store = factor(train$Store)

cat ("There are ", length(levels(train$Store)),"stores/levels in this dataset.\n") ## There are 1000 stores/levels in this dataset. ### Libraries that fail to exploit the linear search strategy: #### The tree package library(tree, quietly=TRUE) try({fit = tree(Sales ~ Store, data = train)}) #### The party package library(party, quietly=TRUE) try({fit = ctree(Sales ~ Store, data = train)}) detach("package:party", unload=TRUE) ## Warning: Error in matrix(0, nrow = mi, ncol = nl) : ## invalid 'nrow' value (too large or NA) ## In addition: Warning message: ## In matrix(0, nrow = mi, ncol = nl) : ## NAs introduced by coercion to integer range #### The partykit package library(partykit, quietly=TRUE) try({fit = lmtree(Sales ~ Store, data = train)}) detach("package:partykit", unload=TRUE) ## Warning: Error in matrix(0, nrow = mi, ncol = nl) : ## invalid 'nrow' value (too large or NA) ## In addition: Warning message: ## In matrix(0, nrow = mi, ncol = nl) : ## NAs introduced by coercion to integer range #### The randomForest package library(randomForest, quietly=TRUE) ## randomForest 4.6-12 ## Type rfNews() to see new features/changes/bug fixes. try({fit = randomForest(Sales ~ Store, data = train, ntree=1)}) #### base lm runs out of memory try({fit0 = lm(Sales ~ Store, data = train)}) ## Warning: Cannot allocate vector of size... ### Libraries that shine: #### The rpart package library(rpart, quietly=TRUE) fit = rpart(Sales ~ Store, data = train) #### The gbm package Well, almost. At least the max number of levels is very high (1024): library(gbm, quietly=TRUE) ## Loaded gbm 2.1.1 fit = gbm(Sales ~ Store, data = train, interaction.depth = 8, n.trees=1) ## Distribution not specified, assuming gaussian ... (Not clear to me why there is a limit at all) #### The RWeka package library(RWeka, quietly=TRUE) #handles only classification problems: train$Sales2 = train$Sales > mean(train$Sales)
fit = J48(Sales2 ~ Store, data = train)

#### The h2o package

library(h2o, quietly=TRUE)
h2o.init(nthreads = -1) #Number of threads -1 means use all cores on your machine

#upload the data to the h2o server
trainHex<-as.h2o(train)

rf_fit1 <- h2o.randomForest(x = "Store",
y = "Sales",
training_frame = trainHex,
model_id = "rf_fit1",
ntrees= 10,
seed = 1)
#plot(rf_fit1)

## One thought on “Categorical Variables in Trees I”

1. Markus, I was just pointed to your blog through the following discussion on SO:

https://stackoverflow.com/questions/44455090/how-to-deal-with-large-number-of-factors-categories-within-partykit

And we had discussed this privately before: The main reason for “partykit” not to implement this shortcut is that it is only available for certain special cases of the very general frameworks provided by ctree() and mob(). We’re currently working on modularizing our internal partykit infrastructure so that such special cases can – in principle – be added. We need more work though for which models/scores/loss functions the trick actually works and where it fails. This is not at the top of our to-do list but at least it’s on it 😉


text:
Réunion d'été SMC 2014

Université du Manitoba, 6 - 9 juin 2014

Sujets variés en théorie de l'approximation
Org: Kirill Kopotun et Andriy Prymak (Manitoba)
[PDF]

ANDRIY BONDARENKO, KNU, Kyiv, Ukraine and NTNU, Trondheim, Norway
On Borsuk's conjecture  [PDF]

In 1933 Karol Borsuk conjectured that every closed set in ${\mathbb R}^n$ of diameter 1 could be partitioned into $n+1$ parts of smaller diameters. The conjecture was disproved in 1993 by Kahn and Kalai. In particular, their construction gives counterexamples for Borsuk's conjecture for $n=1325$ and for all $n>2014$. Until recently the best known result was that Borsuk's conjecture is false for all $n\ge 298$.

We will show how to use the Euclidean representation of strongly regular graphs to construct a two-distance set consisting of $416$ points on the unit sphere in the dimension $65$ which cannot be partitioned into $83$ parts of smaller diameter.

ZEEV DITZIAN, University of Alberta
Measure of smoothness on the unit ball  [PDF]

The classic measure of smoothness of functions on a domain $D$ with respect to a norm as applied to the unit ball is not correct'' that is, it does not satisfy matching direct (Jackson) and weak converse inequalities with respect to algebraic polynomial approximation. To deal with the above deficiency F. Dai and Y. Xu introduced a correct'' measure of smoothness on $L_p(B)$.

Here I am introducing a new different measure of smoothness of functions on the unit ball $B$. This measure beside being correct'' has the added advantages of being rotation invariant (independent of the basis of $R^d$), being applicable to more norms, and being amenable to more applications. Computability of the new measure was demonstrated and some of the computations show optimality of results.

GERMAN DZYUBENKO, Yu.A.Mitropolskiy International Mathematical Center of NAS of Ukraine
Nearly comonotone approximation of periodic functions  [PDF]

Suppose that a continuous $2\pi$-periodic function $f$ on the real axis changes its monotonicity at points $y_i:\, -\pi\le y_{2s}<y_{2s-1}<...<y_1<\pi,\ s\in\Bbb N,$ on each period. In our resent work, for each $n\ge N,$ a trigonometric polynomial $P_n$ of order $cn$ is found such that: $P_n$ changes its monotonicity, like $f,$ everywhere except small intervals $$(y_i-\pi/n,y_i+\pi/n)$$ and $$\|f-P_n\|\le c(s)\,\omega_3(f,\pi/n),$$ where $N$ is a constant depending only on $\min\limits_{i=1,...,2s}\{y_i-y_{i+1}\},\ c$ and $c(s)$ are constants depending only on $s,\ \omega_3(f,\cdot)$ is the modulus of continuity of the $3$-rd order of the function $f,$ and $\|\cdot\|$ is the max-norm.

OLEKSIY KLURMAN, Université de Montréal
Weighted Bernstein and Markov-Nikolskii type inequalities for monotone polynomials of order k  [PDF]

For $n\ge m\ge 0,$ we denote $$M_{q,p}(n,m) :=\sup_{P_n\in\mathbb{P}_n}\frac{\|P^{(m)}_n\|_{L_q [-1,1]}}{\|P_n\|_{L_p [-1,1]}},$$ where $\mathbb{P}_n$ denotes the set of all algebraic polynomials of degree $\le n.$ The asymptotic behaviour of $M_{q,p}(n,m)$ was studied by many authors.

In $1926,$ S. Bernstein showed that $\sup_{P_n\in\triangle_ n}\frac{\|P'_n\|}{\|P_n\|}=\left\{ \begin{array}{ll} \frac{(n+1)^2}{4} , & \mbox{ if } n=2k+1 ,\ \frac{n(n+2)}{4} , & \mbox{ if } n=2k. \end{array} \right.$ where the supremum is taken over all monotone polynomials of degree $\le n.$

Motivated by this result, T. Erdelyi considered analogous problem for $M^{(l)}_{q,p}(n,m),$ where the supremum is taken over all absolutely monotone polynomials of order $l$ and determined the exact asymptotic in the case $p\le q.$

At the time, A. Kroo and J. Szabados found the exact Markov factors for monotone polynomials of order $k$ in $L_1$ and $L_{\infty}$ norms.

In this talk, I will discuss the sharp order of $M^{(l)}_{q,p}(n,m)$ for all values of $p,q.$ I will also discuss the weighted analog of Berstein result for monotone polynomials.

OLEKSANDR MAIZLISH, University of Manitoba

Recently developed adaptive methods, where the hierarchy of triangulations is not fixed in advance and depends on the local properties of the function, have received considerable attention in piecewise polynomial approximation over sets of triangulations. We introduce a new adaptive algorithm which, given a characteristic function of some bounded convex domain (a so-called cartoon image), constructs a hierarchical sequence of triangulations that adapt to the local properties of the function. In case of convex domains with piecewise-smooth boundary, this approximation method implies a theoretically correct" rate of convergence and already outperforms the well-known wavelet methods.

OXANA MOTORNA, Taras Shevchenko National University of Kyiv
On the approximation with Jacobi weights  [PDF]

We will discuss some open problems and results in the the approximation with Jacobi weights $$q_{\alpha,\beta}(x):=(1-x)^\alpha(1+x)^\beta$$ and their generalizations. Say, for the error $$E_n^*(f):=\inf_{P_n}\|(f-P_n)q_{-1/2,-1/2}\|_{C[-1,1]}$$ of the best uniform weighted approximation by algebraic polynomials $P_n$ of degree $<n$ of a continuous on $[-1,1]$ function $f\in \text{Lip} 1$ (that is $|f(x')-f(x'')|\le|x'-x''|, \forall x',x''\in[-1,1]$) the estimate $$nE_n^*(f)\le c$$ is well known. However the exact value of the absolute constant $c$ is not known.

Recently the asymptotically exact estimates for the generalized Lebesgue constants of the Fourier-Jacobi sums in the weighted integral norm are obtained.

For a domain $D\subset \mathbf{C}$ with piecewise smooth boundary an analog of the Jacobi weight is $$q_{\alpha_1,\dots,\alpha_s}(z):=|z_1-z|^{\alpha_1} \dots|z_s-z|^{\alpha_s},$$ where $s$ is the number of the angular points $z_j$. For such domains and weights complete analogs of the estimates, taking place for the interval, are proved.

Finally, we will discuss the shape preserving approximation with Jacobi weights.

ALEXEI SHADRIN, University of Cambridge, UK
Stable reconstruction from Fourier samples  [PDF]

For an analytic and periodic function $f$, the $m$-th partial sums of its Fourier series converge exponentially fast in $m$, but such rapid convergence is destroyed once periodicity is no longer present (because of the Gibbs phenomenon at the end-points).

We can restore higher-order convergence, e.g., by reprojecting the slowly convergent Fourier series onto a suitable basis of orthogonal algebraic polynomials, however, all exponentially convergent methods appear to suffer from some sort of ill-conditioning, whereas methods that recover $f$ in a stable manner have a much slower approximation rate.

We give to these observations a definite explanation in terms of the following fundamental stability barrier: the best possible convergence rate for a stable reconstruction from the first $m$ Fourier coefficients is root-exponential in $m$.

IGOR SHEVCHUK, Taras Shevchenko National University of Kyiv
On the approximation with Jacobi weights  [PDF]

We will discuss some open problems and results in the the approximation with Jacobi weights $$q_{\alpha,\beta}(x):=(1-x)^\alpha(1+x)^\beta$$ and their generalizations. Say, for the error $$E_n^*(f):=\inf_{P_n}\|(f-P_n)q_{-1/2,-1/2}\|_{C[-1,1]}$$ of the best uniform weighted approximation by algebraic polynomials $P_n$ of degree $<n$ of a continuous on $[-1,1]$ function $f\in \text{Lip} 1$ (that is $|f(x')-f(x'')|\le|x'-x''|, \forall x',x''\in[-1,1]$) the estimate $$nE_n^*(f)\le c$$ is well known. However the exact value of the absolute constant $c$ is not known.

Recently the asymptotically exact estimates for the generalized Lebesgue constants of the Fourier-Jacobi sums in the weighted integral norm are obtained.

For a domain $D\subset \mathbf{C}$ with piecewise smooth boundary an analog of the Jacobi weight is $$q_{\alpha_1,\dots,\alpha_s}(z):=|z_1-z|^{\alpha_1} \dots|z_s-z|^{\alpha_s},$$ where $s$ is the number of the angular points $z_j$. For such domains and weights complete analogs of the estimates, taking place for the interval, are proved.

Finally, we will discuss the shape preserving approximation with Jacobi weights.

This is joint work with O. V. Motorna.


text: Hector's banana bread recipe requires $$2 \frac {3}{4}$$ cups of flour, and he has already added $$\frac {2}{3}$$ cups of flour. The amount of flour Hector still needs to add, can be written as $$\frac {a}{b}$$, where $$a$$ and $$b$$ are coprime positive integers. What is the value of $$a+b$$?


text:
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## G = (C2×C20)⋊C8order 320 = 26·5

### 6th semidirect product of C2×C20 and C8 acting via C8/C2=C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — (C2×C20)⋊C8
 Chief series C1 — C5 — C10 — C2×C10 — C2×C20 — C22×C20 — C20.55D4 — (C2×C20)⋊C8
 Lower central C5 — C10 — C2×C10 — (C2×C20)⋊C8
 Upper central C1 — C22 — C22×C4 — C2×C4⋊C4

Generators and relations for (C2×C20)⋊C8
G = < a,b,c | a2=b20=c8=1, ab=ba, cac-1=ab10, cbc-1=ab-1 >

Subgroups: 198 in 78 conjugacy classes, 35 normal (25 characteristic)
C1, C2, C2, C4, C22, C22, C5, C8, C2×C4, C2×C4, C23, C10, C10, C4⋊C4, C2×C8, C22×C4, C20, C2×C10, C2×C10, C22⋊C8, C2×C4⋊C4, C52C8, C2×C20, C2×C20, C22×C10, C22.M4(2), C2×C52C8, C5×C4⋊C4, C22×C20, C20.55D4, C10×C4⋊C4, (C2×C20)⋊C8
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, D5, C22⋊C4, C2×C8, M4(2), Dic5, D10, C22⋊C8, C23⋊C4, C4.10D4, C52C8, C2×Dic5, C5⋊D4, C22.M4(2), C2×C52C8, C4.Dic5, C23.D5, C20.55D4, C23⋊Dic5, C20.10D4, (C2×C20)⋊C8

Smallest permutation representation of (C2×C20)⋊C8
On 160 points
Generators in S160
(61 71)(62 72)(63 73)(64 74)(65 75)(66 76)(67 77)(68 78)(69 79)(70 80)(81 91)(82 92)(83 93)(84 94)(85 95)(86 96)(87 97)(88 98)(89 99)(90 100)(101 111)(102 112)(103 113)(104 114)(105 115)(106 116)(107 117)(108 118)(109 119)(110 120)(121 131)(122 132)(123 133)(124 134)(125 135)(126 136)(127 137)(128 138)(129 139)(130 140)
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20)(21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60)(61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80)(81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100)(101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120)(121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140)(141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160)
(1 98 35 125 144 74 58 108)(2 97 26 134 145 73 49 117)(3 96 37 123 146 72 60 106)(4 95 28 132 147 71 51 115)(5 94 39 121 148 70 42 104)(6 93 30 130 149 69 53 113)(7 92 21 139 150 68 44 102)(8 91 32 128 151 67 55 111)(9 90 23 137 152 66 46 120)(10 89 34 126 153 65 57 109)(11 88 25 135 154 64 48 118)(12 87 36 124 155 63 59 107)(13 86 27 133 156 62 50 116)(14 85 38 122 157 61 41 105)(15 84 29 131 158 80 52 114)(16 83 40 140 159 79 43 103)(17 82 31 129 160 78 54 112)(18 81 22 138 141 77 45 101)(19 100 33 127 142 76 56 110)(20 99 24 136 143 75 47 119)

G:=sub<Sym(160)| (61,71)(62,72)(63,73)(64,74)(65,75)(66,76)(67,77)(68,78)(69,79)(70,80)(81,91)(82,92)(83,93)(84,94)(85,95)(86,96)(87,97)(88,98)(89,99)(90,100)(101,111)(102,112)(103,113)(104,114)(105,115)(106,116)(107,117)(108,118)(109,119)(110,120)(121,131)(122,132)(123,133)(124,134)(125,135)(126,136)(127,137)(128,138)(129,139)(130,140), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20)(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100)(101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140)(141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160), (1,98,35,125,144,74,58,108)(2,97,26,134,145,73,49,117)(3,96,37,123,146,72,60,106)(4,95,28,132,147,71,51,115)(5,94,39,121,148,70,42,104)(6,93,30,130,149,69,53,113)(7,92,21,139,150,68,44,102)(8,91,32,128,151,67,55,111)(9,90,23,137,152,66,46,120)(10,89,34,126,153,65,57,109)(11,88,25,135,154,64,48,118)(12,87,36,124,155,63,59,107)(13,86,27,133,156,62,50,116)(14,85,38,122,157,61,41,105)(15,84,29,131,158,80,52,114)(16,83,40,140,159,79,43,103)(17,82,31,129,160,78,54,112)(18,81,22,138,141,77,45,101)(19,100,33,127,142,76,56,110)(20,99,24,136,143,75,47,119)>;

G:=Group( (61,71)(62,72)(63,73)(64,74)(65,75)(66,76)(67,77)(68,78)(69,79)(70,80)(81,91)(82,92)(83,93)(84,94)(85,95)(86,96)(87,97)(88,98)(89,99)(90,100)(101,111)(102,112)(103,113)(104,114)(105,115)(106,116)(107,117)(108,118)(109,119)(110,120)(121,131)(122,132)(123,133)(124,134)(125,135)(126,136)(127,137)(128,138)(129,139)(130,140), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20)(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100)(101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140)(141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160), (1,98,35,125,144,74,58,108)(2,97,26,134,145,73,49,117)(3,96,37,123,146,72,60,106)(4,95,28,132,147,71,51,115)(5,94,39,121,148,70,42,104)(6,93,30,130,149,69,53,113)(7,92,21,139,150,68,44,102)(8,91,32,128,151,67,55,111)(9,90,23,137,152,66,46,120)(10,89,34,126,153,65,57,109)(11,88,25,135,154,64,48,118)(12,87,36,124,155,63,59,107)(13,86,27,133,156,62,50,116)(14,85,38,122,157,61,41,105)(15,84,29,131,158,80,52,114)(16,83,40,140,159,79,43,103)(17,82,31,129,160,78,54,112)(18,81,22,138,141,77,45,101)(19,100,33,127,142,76,56,110)(20,99,24,136,143,75,47,119) );

G=PermutationGroup([[(61,71),(62,72),(63,73),(64,74),(65,75),(66,76),(67,77),(68,78),(69,79),(70,80),(81,91),(82,92),(83,93),(84,94),(85,95),(86,96),(87,97),(88,98),(89,99),(90,100),(101,111),(102,112),(103,113),(104,114),(105,115),(106,116),(107,117),(108,118),(109,119),(110,120),(121,131),(122,132),(123,133),(124,134),(125,135),(126,136),(127,137),(128,138),(129,139),(130,140)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20),(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60),(61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80),(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100),(101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120),(121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140),(141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160)], [(1,98,35,125,144,74,58,108),(2,97,26,134,145,73,49,117),(3,96,37,123,146,72,60,106),(4,95,28,132,147,71,51,115),(5,94,39,121,148,70,42,104),(6,93,30,130,149,69,53,113),(7,92,21,139,150,68,44,102),(8,91,32,128,151,67,55,111),(9,90,23,137,152,66,46,120),(10,89,34,126,153,65,57,109),(11,88,25,135,154,64,48,118),(12,87,36,124,155,63,59,107),(13,86,27,133,156,62,50,116),(14,85,38,122,157,61,41,105),(15,84,29,131,158,80,52,114),(16,83,40,140,159,79,43,103),(17,82,31,129,160,78,54,112),(18,81,22,138,141,77,45,101),(19,100,33,127,142,76,56,110),(20,99,24,136,143,75,47,119)]])

62 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G 4H 5A 5B 8A ··· 8H 10A ··· 10N 20A ··· 20X order 1 2 2 2 2 2 4 4 4 4 4 4 4 4 5 5 8 ··· 8 10 ··· 10 20 ··· 20 size 1 1 1 1 2 2 2 2 2 2 4 4 4 4 2 2 20 ··· 20 2 ··· 2 4 ··· 4

62 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + - + + - image C1 C2 C2 C4 C8 D4 D5 M4(2) Dic5 D10 C5⋊2C8 C5⋊D4 C4.Dic5 C23⋊C4 C4.10D4 C23⋊Dic5 C20.10D4 kernel (C2×C20)⋊C8 C20.55D4 C10×C4⋊C4 C22×C20 C2×C20 C2×C20 C2×C4⋊C4 C2×C10 C22×C4 C22×C4 C2×C4 C2×C4 C22 C10 C10 C2 C2 # reps 1 2 1 4 8 2 2 2 4 2 8 8 8 1 1 4 4

Matrix representation of (C2×C20)⋊C8 in GL6(𝔽41)

 40 0 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 31 0 0 0 1 0 31 0 0 0 0 40 0 0 0 0 0 0 40
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 36 29 30 33 0 0 12 5 30 27 0 0 0 0 35 6 0 0 0 0 6 6
,
 0 38 0 0 0 0 3 0 0 0 0 0 0 0 31 0 1 29 0 0 31 0 0 29 0 0 40 1 0 0 0 0 39 0 0 10

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,31,31,0,40],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,36,12,0,0,0,0,29,5,0,0,0,0,30,30,35,6,0,0,33,27,6,6],[0,3,0,0,0,0,38,0,0,0,0,0,0,0,31,31,40,39,0,0,0,0,1,0,0,0,1,0,0,0,0,0,29,29,0,10] >;

(C2×C20)⋊C8 in GAP, Magma, Sage, TeX

(C_2\times C_{20})\rtimes C_8
% in TeX

G:=Group("(C2xC20):C8");
// GroupNames label

G:=SmallGroup(320,86);
// by ID

G=gap.SmallGroup(320,86);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,232,387,100,1123,12550]);
// Polycyclic

G:=Group<a,b,c|a^2=b^20=c^8=1,a*b=b*a,c*a*c^-1=a*b^10,c*b*c^-1=a*b^-1>;
// generators/relations

׿
×
𝔽


text:
# Cylindric function

Cylindric function (or cylinder finction or cylindrical function) is class of special functions $$f$$ satisfying equation

$$\!\!\!\!\!\!\!\!\!\!\! (1) ~ ~ ~ z^2 f''(z) + z f'(z) + (z^2-\nu^2) f(z)=0$$

for some fixed value of $$\nu$$ [1]. Often, it is assumed that $$\nu$$ is complex, of real or integer.

This article competes with Bessel function referring to the same topic; and, for summer of 2012, the Bessel function seems to be more advanced. Tho articles on the same subject are kept in TORI in order to elaborate the best terminology: should the term Bessel function refer to only BesselJ, or also to other solutions of the similar equation and include the Neumann function BesselY?

The function with modified argument (multiplied by $$\mathrm i$$ is called Modified cylindric function. The most used are BesseLi and BesselK. In the name BesseLi, the capitalization of the last two letters is swapped because many font designers are lazy and did not provide different images for characlters I, l and | ; the notation suggested allows to avoid confusions: L and i are easier to distinguish than l and I.

Several cylindric functions have special names: BessleJ = J, BesselY=Y, BesselH=H, or, correspondently, Bessel function, Neumann function and Hankel function. Short names with single letters are also used; the corresponding value of parameter $$\nu$$ in equation (1) is indicated as subscript. These functions are related with equation

$$\!\!\!\!\!\!\!\!\!\!\! (2) ~ ~ ~ H_\nu(z)=J_\nu(z)+\mathrm i Y_\nu(z)$$

The term Bessel function is often applied also to all cylindric functions [2]. In this sense, for example, BesselY is also Bessel function, although it has more specific name Neumann function. In order to keep the name Bessel function for BessleJ, the term Cylindric function is used in this article. The modified cylindric functions also are considered as cylindric functions.

Equatoin (1) and cylindric functions often appear in the consideration of waves in a system with circular symmetry; in particular, functions BesselJ0 and BesselK0 are used to build the principal mode of an idealized circular waveguide.

## References

1. http://encyclopedia2.thefreedictionary.com/Cylindrical+Function Cylindrical Function. Warning: The following article is from The Great Soviet Encyclopedia (1979)...


text:
# Definabilty of two functions on natural numbers

Is there a first order logic arithmetic ($0,+,\cdot$) formula $f(n,m)$ such that $f(n,m)$ is true in $\mathbb{N}$ iff $m$ is the $n$th prime number? Similarly, is there $g(k,n,m)$ which is equivalent to $k^n=m$ ?

I have no clue, but I suppose this could potentially be difficult to answer.

-

I assume that you mean the signature $(0, +, \cdot, =)$, so that there is at least one relation symbol. The answer is that the things in the question are definable, and this is well known.

First, let's recover the language of Peano arithmetic. In the signature of the question, we can define:

• the order relations: $x < y$ if and only if $x \not = y$ and $(\exists z)[z+x=y]$
• the successor function: $x = S(y)$ if and only if $x < y$ and $\lnot (\exists z)[x < z \land z < y]$
• the number $1$, which is $S(0)$.

So we may as well assume we have the "full" signature of Peano arithmetic, $(0, 1, S, +, \cdot, <, =)$.

The collection of functions and relations definable on the natural numbers with this signature is described by the arithmetical hierarchy. In particular, it includes all computable functions on the natural numbers. Both of the functions/relations mentioned in the question are computable, so they are definable. But many non-computable functions and relations are also definable.

The main technical step in showing that all computable functions can be defined is showing that it is possible to quantify over finite sequences of natural numbers. One standard way of doing this goes through the β function developed by Gödel for this purpose.

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Yes there is. For every recursive function $f(x_1,\dots,x_n)$, there is a formula $\varphi(y,x_1,\dots,x_n)$ of arithmetic such that for any non-negative integers $b,a_1,\dots,a_n$, we have

(i) If $f(a_1,\dots,a_n)=b$, then the formula $\varphi(b,a_1,\dots,a_n)$ is provable in (say) first-order Peano Arithmetic.

(ii) If $f(a_1,\dots,a_n)\ne b$, the negation of $\varphi(b,a_1,\dots,a_n)$ is provable in (say) first-order Peano Arithmetic.

First-order Peano Arithmetic can be replaced here by substantially weaker theories.

The formula $\varphi$ is called a representing formula for $f$,

I have been a little sloppy. We cannot put numbers into formulas. So in $\varphi(b,a_1,\dots,a_n)$, we should, if for example $b=3$, replace $b$ by the formal term $(1+1)+1$.

The representability result is crucial for the proof that first-order Peano arithmetic is undecidable.

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Oh, alright. I have to read into that.. –  user35359 Feb 3 '13 at 12:49