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## Introduction

Nitrification, the microbially mediated oxidation of ammonia (NH3) to nitrate (NO3) via nitrite (NO2), is a key process of the biogeochemical nitrogen cycle [1, 2] and is mostly driven by autotrophic microorganisms that are capable of growing with NH3 and/or NO2 as sole energy generating substrates. For more than a century, ammonia-oxidizing bacteria (AOB) were considered the lone drivers of aerobic ammonia oxidation by autotrophs, as ammonia-oxidizing archaea (AOA) [3, 4] and complete ammonia oxidizers (comammox) [5,6,7] eluded discovery until relatively recently. Our present-day understanding of ammonia oxidation is quite different: AOA frequently outnumber AOB in oligotrophic habitats [8,9,10], while AOB often dominate in eutrophic environments [11,12,13,14]. Comammox have been shown to be abundant and sometimes even dominant in various natural and engineered environments [15,16,17,18,19], although the habitat range and ecophysiology of comammox remains less well resolved. Notably, in the majority of ecosystems—with the exception of the marine environment, where no comammox has been detected—AOA, AOB, and comammox often co-occur.

Many environmental and physiological factors are known to affect the niche differentiation and habitat selection of ammonia-oxidizing microorganisms (AOM) [20, 21]. In fact, AOM species display differential responses to factors such as pH, oxygen concentrations, light conditions, temperature, metal and organic compounds, and substrate concentrations [22,23,24,25,26,27]. These differential responses are frequently used to explain the co-occurence of AOM across environments. However, the cellular properties underlying these niche-differentiating physiological characterstics of AOM often remain unclear.

The substrate affinity of a microorganism can be expressed with Michaelis–Menten kinetic equations, analogous to enzyme kinetics, defined by an apparent-half-saturation concentration (substrate affinity; Km(app)) and a maximal reaction rate (Vmax). In addition, the specific substrate affinity (a°; Vmax divided by Km(app)) takes into account both the cellular Km(app) and Vmax, and is thus an appropriate measure for comparing interspecies competitiveness [28]. Throughout this study, instantaneous substrate-dependent kinetic properties were measured which represent cellular activity in the absence of growth. Therefore, the Michaelis–Menten modeled kinetic properties discussed here (Km(app) and Vmax) differ from Monod modeled kinetic properties (Ks and μmax), which also take into account other cellular process (e.g., growth, division, stress, and repair). Our focus on Michaelis–Menten modeled kinetic properties is in part due to the enormous challenges that would be associated with growing such a wide array of AOA under the continuous cultivation conditions necessary to determine Monod kinetic properties.

Based on whole cell kinetic properties, AOM were observed to have different survival or lifestyle strategies. The first study investigating the whole cell kinetics of an AOA revealed that Nitrosopumilus maritimus SCM1 displayed a low maximum NH3 oxidation rate (Vmax), but a very high substrate affinity and a° for NH3, compared with AOB [29]. Based on these findings with a single AOA strain, substrate affinity was postulated as a major niche-differentiating factor between AOA and AOB [20, 29]. However, recently it was shown that (i) the only comammox isolate Nitrospira inopinata has a Km(app) for NH3 lower than that of all characterized AOB and (ii) that the Km(app) for NH3 in a few non-marine AOA strains is not always orders of magnitude lower than that of AOB [5]. Nevertheless, the AOA with comparatively high Km(app) for NH3 (low affinity) still possess a significantly higher a° than AOB, indicating that these AOA are still more efficient substrate scavengers [5]. Furthermore, temperature and pH, which are known niche-differentiating factors [30,31,32], have previously been shown to affect the substrate affinity of AOB [33,34,35], but the influence of these parameters on the substrate affinity of AOA and comammox remains to be determined.

In this study, the whole cell kinetic properties of 12 AOA species were determined through instantaneous substrate-dependent microrespirometry (MR) experiments. These include representatives from all four major AOA phylogenetic lineages, isolated or enriched from various habitats (i.e., marine, terrestrial, and geothermal) and possessing a wide variety of pH and temperature growth optima. In these analyses, we also explored the links between the cellular Km(app) and a° of AOM with their cell surface area to volume (SA/V) ratio. Furthermore, by performing MR experiments at different pH values we investigated whether the undissociated NH3 or ammonium (NH4+) is the substrate for AOA and comammox.

## Materials and methods

### Cultivation of ammonia oxidizers

Several previously described growth media were used to cultivate the AOM used in this study. A comprehensive guide with medium components and cultivation conditions is provided in the Supplementary Materials and Methods, Tables S1, and S2. Briefly, all cultures were grown without shaking, in the dark, at their optimum growth temperature and pH, unless otherwise stated. Ammonium (NH4Cl) from pre-sterilized stocks was added as substrate as needed. The growth medium of Nitrosarchaeum koreense MY1, ‘Ca. Nitrosotenuis chungbukensis’ MY2, ‘Ca. Nitrosotenuis uzonensis’ N4, N. maritimus SCM1, Nitrosopumilus piranensis D3C, and Nitrosopumilus adriaticus NF5 was supplemented with sodium pyruvate (0.5 mM) at all times. The pH of all growth media were adjusted when necessary by addition of sterile NaHCO3. Ammonia oxidation activity was determined by measuring ammonium, nitrite, and nitrate concentrations photometrically [36,37,38,39] using an Infinite 200 Pro M Nano+ spectrophotometer (Tecan Group AG, Switzerland).

### Novel AOA enrichments and pure culture

The sampling site, enrichment process, and initial strain characterization details for the two novel thermophilic AOA enrichment cultures ‘Ca. Nitrosofervidus tenchongensis’ DRC1, and ‘Ca. Nitrososphaera nevadensis’ GerE used in this study are provided in the Supplementary Materials and Methods. In addition, in this study, ‘Ca. N. uzonensis’ N4 was isolated as a pure culture from a previously described geothermal spring enrichment culture [40]. Further details are provided below and in the Supplementary Materials and Methods.

### Phylogenetic analysis

Taxa chosen for phylogenetic reconstruction met at least one of three different criteria: (1) inclusion in the current study, (2) published as a culture/enrichment or (3) designation as a “GTDB representative of species” for the group of taxa classified as “Nitrososphaerales” by the Genome Taxonomy Database release 05-RS95 [41]. A concatenated alignment of 34 universal genes (43 markers) was automatically constructed using CheckM [42]. IQ-TREE v 2.1.2 [43] was used for phylogenetic reconstruction following automatic model determination with modelFinder [44], which identified LG + F + R6 as the best-fit model according to the Bayesian Information Criterion (BIC). Bipartition support was determined with ultrafast bootstraps (UFboots [45]).

### Substrate-dependent oxygen uptake measurements

Cellular substrate oxidation kinetics were determined from instantaneous substrate-dependent oxygen uptake measurements as previously described [5, 29, 46]. Briefly, measurements were performed with a MR system, equipped with a PA 2000 picoammeter and a 500 μm tip diameter OX-MR oxygen microsensor (Unisense, Denmark), polarized continuously for at least 24 h before use.

Active AOA, AOB, and N. inopinata cells were harvested (4000 × g, 10 min, 20 °C) from ammonium replete active cultures, using 10 kDa-cutoff, Amicon Ultra-15 centrifugal filter units (Merck Millipore, Germany). Concentrated cells were washed with and resuspended in substrate-free medium appropriate for the respective cultures. Exceptions were ‘Ca. Nitrosocosmicus franklandus’ C13 and the marine AOA, N. maritimus SCM1, N. piranensis D3C, and N. adriaticus NF5. These four AOA strains were not active in the MR chambers after attempts to concentrate their biomass. Therefore, ammonium concentrations were monitored daily for these four cultures, and cells were used without concentration for MR promptly upon substrate depletion [29]. AOM harvested cells for MR experiments were incubated for at least 30 min in a recirculating water bath set to the experimental temperature (Tables S2 and S3) prior to being transferred to the MR chambers (~2 ml).

In addition to MR experiments at optimal growth temperature and pH (Table S2), MR experiments were also performed at non-optimal growth temperatures and medium pH (Table S3). ‘Ca. N. oleophilus’ MY3 was cultivated at 30 °C, harvested with centrifugal filter units (see above), and incubated for ~2 h in substrate-free medium across a range of temperatures (25, 30, and 35 °C). MR experiments were then performed at the respective preincubation temperature. Likewise, N. inopinata and ‘Ca. N. oleophilus’ MY3 cells were harvested with centrifugal filter units (see above) and resuspended in substrate-free medium containing 10 mM HEPES (pH 7.4). The pH was adjusted to 6.5–8.4 with 1 M HCl or 1 M NaOH (Table S3). These cultures were then incubated at their optimum growth temperature for ~1 h prior to cellular kinetic measurements. Culture pH was determined before and after oxygen uptake measurements to confirm the pH did not change during MR. Substrate-dependent oxygen uptake measurements were performed as described below.

For all MR experiments, glass MR chambers containing glass-coated magnetic stir bars were filled headspace-free, sealed with MR injection lids, and submerged in a recirculating water bath. An OX-MR microsensor was inserted into each MR chamber and left to equilibrate (300 rpm, ~1 h). Exact temperatures used for each culture and experiment are provided in Tables S2 and S3. Stable background sensor signal drift was measured for at least 15 min prior to initial substrate injections, and the background oxygen consumption rate was subtracted from the measured oxygen uptake rates. Hamilton syringes (10 or 50 μl; Hamilton, USA) were used to inject NH4Cl stock solutions into MR chambers. Both single and multiple trace oxygen uptake measurements were performed. For single trace measurements, a single substrate injection was performed, and oxygen uptake was recorded until substrate depletion. For multiple trace measurements, multiple injections of varying substrate concentration were performed in a single MR chamber. Once stable, discrete slopes of oxygen uptake were calculated following each substrate injection. Immediately following oxygen uptake measurements, the total ammonium concentration and pH of the MR chamber contents were determined. The cells were stored at −20 °C for protein analysis. Cells were lysed with the Bacterial Protein Extraction Reagent (BPER, Thermo Scientific) and the total protein content was determined photometrically with the Pierce bicinchoninic acid Protein Assay Kit (Thermo Scientific) as per the manufacturer’s instructions. Before and after MR assays of N. piranensis D3C, ‘Ca. N. uzonensis’ N4, and ‘Ca. N. oleophilus’ MY3, an aliquot of culture was filtered onto membranes (0.2 µm polycarbonate GTTP membranes; Merck Milipore, Germany) and DAPI (4ʹ,6-diamidino-2-phenylindole; 10 µg ml−1; 5 min; room temperature) stained prior to microscopic measurement of biomass cell aggregate size, as described previously [47, 48].

### Calculation of kinetic properties

Km(app) and Vmax were calculated from both single and multiple trace substrate-dependent oxygen uptake measurements. Total ammonium (NH3 + NH4+) oxidation rates were calculated from oxygen uptake measurements using a substrate to oxygen consumption ratio of 1:1.5 [5, 29]. Total ammonium uptake rates were fitted to a Michaelis–Menten model using the equation:

$$V = \left( {V_{\max } \times \left[ S \right]} \right) \times \left( {K_{{{{{{\mathrm{m}}}}}}({{{{{\mathrm{app}}}}}})} + \left[ S \right]} \right)^{ - 1}$$
(1)

where V is the reaction rate (μM h−1), Vmax is the maximum reaction rate (μM h−1), S is the total ammonium concentration (μM), and Km(app) is the reaction half saturation concentration (μM). A nonlinear least squares regression analysis was used to estimate Km(app) and Vmax [49]. The Km(app) for NH3 for each strain was calculated based on the Km(app) for total ammonium, incubation temperature, pH, and salinity [50]. Km(app) values for AOM not determined in this study were compiled from the literature [5, 29, 33, 34, 51,52,53,54,55]. If only total ammonium information was given by the authors for Km(app), the corresponding NH3 values were calculated based on the reported experimental temperature, pH, and salinity values. Vmax values of pure cultures were normalized to culture protein content. The specific substrate affinity (a°; l g wet cells−1 h−1) of each pure culture strain was calculated using the equation:

$$a^o = \left( {\frac{{V_{\max }}}{{{{{{{\mathrm{cellular}}}}}}\;{{{{{\mathrm{protein}}}}}}\;{{{{{\mathrm{x}}}}}}\;5.7}}} \right){{{{{\mathrm{x}}}}}}\,K_{{{{{{\mathrm{m}}}}}}({{{{{\mathrm{app}}}}}})^{ - 1}}$$
(2)

Where the Vmax is normalized to the protein concentration (g l−1) of the culture in the MR chamber and the factor of 5.7 g wet cell weight per g of protein was used for all AOM [5, 29, 56]. The a° for NH3 or total ammonium were calculated using the respective Km(app) for NH3 or total ammonium.

### Cell surface area to volume ratio calculation

Approximate cell SA/V ratios were determined using cell dimensions provided by or calculated from previously published phase contrast, transmission electron, or scanning electron microscopy images (Table S4). The following equations for the surface (SA) area and volume (V) of a sphere (3) and rod (4) were used:

$${{{{{\mathrm{SA}}}}}} = 4\pi {{{{{\mathrm{r}}}}}}^2;\;{{{{{\mathrm{V}}}}}} = 4/3\pi {{{{{\mathrm{r}}}}}}^3$$
(3)
$${{{{{\mathrm{SA}}}}}} = 2\pi {{{{{\mathrm{r}}}}}}({{{{{\mathrm{h}}}}}} + {{{{{\mathrm{r}}}}}});\;{{{{{\mathrm{V}}}}}} = \pi {{{{{\mathrm{r}}}}}}^2{{{{{\mathrm{h}}}}}}$$
(4)

where r is the cell radius (µm) and h is the cell length (µm). The cell size and volume from published phase contrast images were verified using MicrobeTracker [57].

## Results and discussion

### AOA kinetic properties

In this study we investigated the kinetic properties of 12 AOA strains, including representatives from all four described AOA phylogenetic lineages: Nitrosopumilales (Group I.1a), ‘Ca. Nitrosotaleales’ (Group I.1a-associated), Nitrososphaerales (Group I.1b), and ‘Ca. Nitrosocaldales’ (thermophilic AOA clade) [58, 59] (Fig. 1). These AOA isolates and enrichments were obtained from a variety of habitats (marine, soil, sediment, hot spring) and have optimal growth pH and temperatures ranging from 5.3–7.8 to 25–72 °C, respectively (Table S2). The substrate-dependent oxygen consumption rates for all AOA tested followed Michaelis–Menten kinetics. Below, the kinetic properties of these AOA are put into a broader context with comparisons to previously characterized AOM. It is important to note that the whole cell kinetic properties, such as substrate competitiveness, detailed here were generated from instantaneous activity measurements in the absence of growth. It is unknown how the substrate competitiveness of nitrifiers may or may not differ from their competitiveness when cellular processes such as growth, division, stress, and repair are involved.

#### Nitrosopumilales (Group I.1a)

From this lineage, three mesophilic marine (N. piranensis D3C, N. adriaticus NF5, and N. maritimus SCM1) [3, 60], two agricultural soil (N. koreense MY1 and ‘Ca. N. chungbukensis’ MY2) [61, 62] and one thermal spring isolate (‘Ca. N. uzonensis’ N4) [40] were kinetically characterized (Fig. S1). These AOA all displayed a high substrate affinity for NH3, ranging from ~2.2 to 24.8 nM. Thus, all characterized Nitrosopumilales, and not just marine isolates, are adapted to oligotrophic conditions. All possess substrate affinities several orders of magnitude higher (lower Km(app)) than any characterized AOB, with the exception of the recently characterized acidophilic gammaproteobacterial AOB ‘Ca. Nitrosacidococcus tergens’ [55] (Fig. 2a). This finding appears to support the widely reported hypothesis that regardless of the environment, AOA in general are adapted to lower substrate concentrations than AOB [22, 29, 30]. However, as described later, this trend does not apply to all AOA.

As the substrate oxidation kinetics of the marine AOA strain, N. maritimus SCM1, originally characterized by Martens-Habbena et al. [29] have recently been disputed [63], they were revisited in this study (Fig. S2). With the same strain of N. maritimus used in Hink et al. [63] (directly obtained by the authors), we were able to reproduce (Figs. S1 and S2) the original kinetic properties of N. maritimus SCM1 reported in Martens-Habbena et al. [29] ruling out strain domestication during lab propagation as cause for the observed discrepancy. Therefore, the reported differences in the literature possibly reflect the measurements of two distinct cellular properties, Km(app) [29] and Ks [63], representing the half saturation of activity and growth, respectively. In addition, differences in pre-measurement cultivation and growth conditions could also contribute to these unexpected differences [63, 64]. More details are provided in the Supplementary Results and Discussion.

#### ‘Ca. Nitrosotaleales’ (Group I.1a-associated)

The only isolated AOA strains in this lineage ‘Ca. Nitrosotalea devanaterra’ Nd1 and ‘Ca. Nitrosotalea sinensis’ Nd2, are highly adapted for survival in acidic environments and grow optimally at pH 5.3 [25, 65]. Both display a relatively low affinity for total ammonium (Km(app) = 3.41–11.23 μM), but their affinity for NH3 is among the highest calculated of any AOA characterized (Km(app) = ~0.6–2.8 nM) (Fig. 2a,c, and Fig. S3). This seemingly drastic difference in substrate affinity for total ammonium versus NH3 is due to the combination of the high acid dissociation constant of ammonium (pKa = 9.25) and the kinetic properties of these strains being carried out at pH 5.3. The very limited availability of NH3 under acidic conditions has led to the hypothesis that these acidophilic AOA should be highly adapted to very low NH3 concentrations and possess a high substrate affinity (low Km(app)) for NH3 [66, 67]. Our data corroborate this hypothesis.

#### Nitrososphaerales (Group I.1b)

The AOA strains ‘Ca. N. nevadensis’ GerE (culture information provided in Supplementary Results and Discussion), ‘Ca. N. oleophilus’ MY3 [68] and ‘Ca. N. franklandus’ C13 [69] were kinetically characterized, and contextualized with the previously published kinetic characterization of Nitrososphaera viennensis EN76 and ‘Ca. Nitrososphaera gargensis’ [5]. Together, the Nitrososphaerales AOA possess a wide range of affinities for NH3 (Km(app) = ~0.14–31.5 µM) (Fig. 2a and Fig. S4). Although this range of NH3 affinities spans more than two orders of magnitude, none of the Nitrososphaerales AOA possess an affinity for NH3 as high as any Nitrosopumilales or ‘Ca. Nitrosotaleales AOA (Fig. 2a).

The moderately thermophilic enrichment culture ‘Ca. N. nevadensis’ GerE displayed a higher substrate affinity (lower Km(app)) for NH3 (0.17 ± 0.03 µM) than the other characterized AOA strains within the genus Nitrososphaera (Fig. 2a). In contrast, ‘Ca. N. oleophilus’ MY3 and ‘Ca. N. franklandus’ C13, which belong to the genus Nitrosocosmicus, had the lowest affinity (highest Km(app)) for NH3 (12.37 ± 6.78 μM and 16.32 ± 14.11 μM, respectively) of any AOA characterized to date. In fact, their substrate affinity is comparable to several characterized AOB (Fig. 2a). In this context it is interesting to note that several Nitrosocosmicus species have been shown to tolerate very high ammonium concentrations [68,69,70], a trait usually associated with AOB [24, 54]. The low substrate affinity observed in Nitrosocosmicus AOA correlates with the absence of a putative Amt-type high affinity ammonium transporter in the genome of any sequenced Nitrosocosmicus species to date [68, 69, 71].

#### ‘Ca. Nitrosocaldales’ (Thermophilic AOA lineage)

The thermophilic AOA enrichment cultures, ‘Ca. Nitrosocaldus yellowstonensis’ HL72 [72] and ‘Ca. N. tenchongensis’ DRC1 (culture information provided in Supplementary Results and Discussion), possess affinities for NH3 (Km(app) = ~1.36 ± 0.53 μM and ~0.83 ± 0.01 μM; respectively comparable to AOA within the genus Nitrososphaera (Fig. 2a). Notably, the substrate oxidation rate of these two AOA quickly dropped with increasing substrate concentrations after Vmax was reached (Fig. S5). This trend was not observed with any other AOA tested here and may reflect an increased susceptibly to NH3 stress at high temperatures, as the free NH3 concentration increases with increasing temperatures [33]. It should be noted that both of these AOA cultures are enrichment cultures, as no member of the ‘Ca. Nitrosocaldales’ has been isolated to date.

Together, these results highlight that the substrate affinity for NH3 among AOA species is much more variable than previously hypothesized, spanning several orders of magnitude and in some cases overlapping with the substrate affinity values of characterized non-oligotrophic AOB. In addition, the substrate affinity of AOA is related, to a certain degree, to their phylogenetic placement within each of the four AOA phylogenetic lineages mentioned above (Fig. 2). Although the substrate affinity ranges of these AOA lineages overlap, the link between AOA phylogeny and kinetic properties provides deeper insights into the physiological and evolutionary differences among AOA species. As a limited number of AOA have been isolated and characterized to date, the continued isolation and characterization of AOA from underrepresented phylogenetic lineages and new habitats is needed. While substrate affinity is certainly one of multiple factors that contribute to niche differentiation between AOM in general, it may also present a previously under acknowledged factor in AOA niche differentiation.

#### Maximum substrate oxidation rates (Vmax)

The normalized maximum substrate oxidation rate of all the AOA characterized to date only span about one order of magnitude from 4.27 to 54.68 μmol N mg protein−1 h−1. These normalized AOA Vmax values are in the same range as the recorded Vmax for the comammox N. inopinata (~12 μmol N mg protein−1 h−1) and the marine AOB strain Nitrosococcus oceani ATCC 19707 (~38 μmol N mg protein−1 h−1) but are lower than the normalized Vmax of the AOB Nitrosomonas europaea ATCC 19718 (average of 84.2 μmol N mg protein−1 h−1; Fig. 2e). The high Vmax value for N. europaea is the only real outlier among the AOM characterized to date and it remains to be determined whether other AOB related to N. europaea also possess such a high Vmax or if members of the Nitrosomonadales possess a broad range of Vmax values. Similarly, as additional comammox strains become available as pure cultures their kinetic characterization will be vital in understanding the variability of these ecologically important parameters within this guild.

#### Specific substrate affinity (a°)

Although the Km(app) and Vmax of AOM can be compared by themselves and provide useful information on cellular properties, the ability of an AOM to scavenge (and compete for) substrate from a dilute solution is most appropriately represented by the a°, which takes into account both the cellular Km(app) and Vmax [28]. In previous studies, the a° of AOM has been calculated using the Km(app) value for total ammonium (NH3 + NH4+) and not the Km(app) value for NH3 [5, 29]. Calculating the a° based on the Km(app) value for total ammonium allows for the a° of AOM to be compared with the a° of microorganisms that do not use NH3 as a sole energy generating substrate, such as ammonium assimilating heterotrophic bacteria or diatoms [29]. While this is useful when evaluating competition for total ammonium in mixed communities or environmental settings, an a° calculated using the Km(app) value for NH3 may be more useful when directly comparing the interspecies competitiveness of AOM for the following reasons: (i) our data support the hypothesis that the substrate for all AOM is NH3 and not NH4+ (see below) and (ii) the Km(app) value for total ammonium is more dependent on the environmental factors it was measured at (e.g., pH, temperature, salinity) than the Km(app) for NH3.

All characterized AOA (with the exception of representatives of the genus Nitrosocosmicus) and the comammox bacterium N. inopinata possess much higher a° for total ammonium or NH3 (~10–3000×) than the AOB, N. oceani or N. europaea (Fig. 2b–d), indicating that they are highly competitive in environments limited in either total ammonium or only NH3. However, due to the low number of published normalized Vmax values for AOB, a° could only be calculated for these two AOB representatives. Thus, extrapolations to the a° of all AOB species, based solely on these observations should be approached with caution.

The low variation in experimentally measured Vmax values (Fig. 2e) across all measured AOM in combination with the high variation in Km(app) values leads to a strong relationship between cellular a° and the reciprocal of Km(app) (Fig. 3) according to Eq. 2 (see Materials and Methods). AOM adapted to oligotrophic (low substrate) conditions should possess both a high substrate affinity (low Km(app)) and a high ao [28]. Therefore, the AOM best suited for environments limited in total ammonium are the AOA belonging to the Nitrosopumilales and the comammox isolate N. inopinata, (top right corner of Fig. 3a). Overall, when looking at solely NH3 or total ammonium, the separation of species in these plots remains similar, with the exception that the acidophilic AOA belonging to the ‘Ca. Nitrosotaleales’ are predicted to be best suited for life in environments limited in NH3 (Fig. 3b). The adaptation correlates well with the fact the AOA ‘Ca. Nitrosotalea devanaterra’ Nd1 and ‘Ca. Nitrosotalea sinensis’ Nd2 were isolated from acidic soils with a pH of 4.5 and 4.7, respectively [25, 65], where the NH3:NH4+ equilibrium is heavily shifted toward NH4+.

In either case, when looking at NH3 or total ammonium, the AOA belonging to the genus Nitrosocosmicus (‘Ca. N. oleophilus’ MY3 and ‘Ca. N. franklandus’ C13) and AOB populate the lower left section of these plots, indicating that they are not strong substrate competitors in NH3 or total ammonium limited environments (Fig. 3). Here, the Vmax of all the AOM reported spans ~10×, whereas the difference in Km(app) spans about five orders of magnitude. If the cellular kinetic property of Vmax really is so similar across all AOB, AOA, and comammox species (Fig. 2e) compared to the large differences in Km(app) values, then substrate competitiveness can be predicted from an AOMs Km(app) for either NH3 or total ammonium (Fig. 2a–c). This may prove especially helpful when characterizing enrichment cultures, where normalizing ammonia-oxidizing activity to cellular protein in order to obtain a comparable Vmax value is not possible. However, there is also a need for more kinetically characterized AOB and comammox species to confirm this hypothesis. In addition, when comparing AOM, differences in the Vmax cellular property will play a larger role, the closer the Km(app) values of the AOM strains are. This is important to consider when comparing AOM from similar habitats and likely adapted to similar substrate concentrations.

#### The effect of environmental and cellular factors on AOA kinetic properties

The concentration of NH3 present in a particular growth medium or environment can vary by orders of magnitude, based solely on the pH, temperature, or salinity of the system [73]. This is notable because at a given total ammonium concentration, the concentration of NH3 is ~10 times higher at 70 °C versus 30 °C and ~1000 times lower at pH 5.3 versus pH 8.4 (representative of maximum ranges tested). While it should be recognized that in our dataset no AOM were included that have a pH optimum between 5.3 and 7.0, the effect of pH and temperature on the ammonia oxidation kinetics of AOM must be considered in order to understand their ecophysiological niches. However, there was no correlation between the kinetic properties of AOM (Km(app), Vmax, and a°) measured in this study and their optimal growth temperature or pH. This lack of correlation between AOM species kinetic properties and growth conditions does not imply that the cellular kinetic properties of an individual AOM species will remain the same over a range of pH and temperature conditions. Therefore, we investigated the effect of pH and temperature variation on the substrate-dependent kinetic properties of the AOA strain ‘Ca. N. oleophilus’ MY3, and the effect of pH on the comammox strain N. inopinata. Here, the AOA ‘Ca. N. oleophilus’ MY3 was selected based on the fact that it is a non-marine, mesophilic, pure culture, that does not require external hydrogen peroxide scavengers for growth. These traits are shared with the previously characterized AOB, N. europaea [35], and the comammox organism, N. inopinata (this study) and thus facilitate comparison.

#### The effect of temperature

The effects of short-term temperature changes on the substrate-dependent kinetic properties of ‘Ca. N. oleophilus’ MY3 were determined. Temperature shifts of 5 °C above and below the optimal growth temperature (30 °C) had no effect on the Km(app) for total ammonium. However, the Km(app) for NH3, Vmax, and a° of ‘Ca. N. oleophilus’ MY3 all increased with increasing temperatures (Fig. S6). Therefore, as temperature increased, ‘Ca. N. oleophilus’ MY3 displayed a lower substrate affinity (higher Km(app) for NH3) but would be able to turnover substrate with a higher Vmax and better compete for substrate with a higher a°. Increasing AOA Km(app) values for NH3 with increasing temperatures have also been observed across studies with N. viennensis EN76 (Fig. S2), and this is discussed in more detail in the Supplementary Results and Discussion. In addition, similar observations have previously been made for AOB strains belonging to the genus Nitrosomonas [33, 34]. The increase in Vmax and a° can be explained in terms of the Van’t Hoff rule (reaction velocity increases with temperature) [74], or in terms of a temperature sensitivity coefficient (Q10; change in reaction velocity over 10 °C) [75]. Here, the maximal reaction velocity of ‘Ca. N. oleophilus’ MY3, displays a relative Q10 of 2.17 between 25 and 35 °C, which is in line with more general microbial respiration measurements [75, 76].

The increase in Km(app) for NH3 (lower NH3 affinity) with increasing temperature is less straightforward to interpret. As this is a whole cell measurement, the observed differences may result from either broad cellular changes or from changes in individual enzymes involved in the ammonia oxidation pathway specifically. At the cellular level, changes in the proteinaceous surface layer (S-layer) or lipid cell membrane could affect substrate movement/transport and enzyme complex stability. It has been suggested that the negatively charged AOA S-layer proteins act as a substrate reservoir, trapping NH4+ and consequently increasing the NH3 concentration in the AOA pseudo-periplasmic space [77]. It is interesting to note that sequenced representatives from the genus ‘Ca. Nitrosocosmicus’ lack the main S-layer protein (slp1) found in all Nitrosopumilales, Nitrososphaerales, and ‘Ca. Nitrosotaleales’ sequenced isolates [71], although it remains to be demonstrated whether ‘Ca. Nitrosocosmicus’ members actually lack a S-layer or form S-layers composed of other proteins. In addition, it has been demonstrated that elevated temperatures significantly alter the lipid composition in the AOA cell membrane [78, 79]. However, it is unclear how differences in the cell membrane or S-layer composition between AOA species may affect the observed kinetic properties. In this context it is important to note that on the single enzyme level, previous studies have shown the same trend of decreasing substrate affinity and increasing maximal reaction velocity with increasing temperatures, due to altered protein structures and an increased enzyme-substrate dissociation constant [80, 81].

Notably, differing optimum growth and activity conditions were previously determined for the marine AOB strain Nitrosomonas cryotolerans [34]. These observations raise interesting, albeit unanswered, questions about why the growth and activity temperature optima are or can be uncoupled in AOM, and what this means for AOM niche differentiation and their competitiveness in-situ. Moving forward, investigations into the growth and cellular kinetic properties of AOM across a range of environmental factor gradients will be essential in understanding competition between AOM in engineered and environmental systems.

#### The effect of pH

The effects of short-term pH changes on the substrate-dependent kinetics of ‘Ca. N. oleophilus’ MY3 and N. inopinata were determined. The Vmax of both ‘Ca. N. oleophilus MY3’ and N. inopinata were stable at 37.3 ± 6.6 μmol N mg protein−1 h−1 and 11.2 ± 2.5 μmol N mg protein−1 h−1, respectively, in medium with a pH between ~6.5 and ~8.5 (Table S3). The Km(app) for total ammonium of ‘Ca. N. oleophilus MY3’ and N. inopinata decreased by more than an order of magnitude (~11×) across this pH range, while the Km(app) for NH3 remained more stable, increasing only 3–4 times (Fig. 4). This stability of the Km(app) for NH3 compared with the larger change in the Km(app) for total ammonium across this pH range suggests that the actual substrate used by AOA and comammox is indeed the undissociated form (NH3) rather than the ammonium ion (NH4+), as previously demonstrated for AOB [34, 35, 54, 82]. As these kinetic measurements were performed with whole cells, the change in Km(app) for NH3 across this pH range may be due to cellular effects of the differing pH values unrelated to the direct ammonia oxidation pathway. The changes in Km(app) for NH3 and Km(app) for total ammonium demonstrated here for ‘Ca. N. oleophilus’ MY3 and N. inopinata are similar to what has been observed for AOB. That AOA and AOB utilize the NH3 as a substrate, aligns with the fact that both are competitively inhibited by the non-polar acetylene compound [83, 84].

Interestingly, the only exception to this rule to date is the gammaproteobacterial marine AOB Nitrosococcus oceani. The reported Km(app) for total ammonium of N. oceani remained more stable (~2.3×) than the Km(app) for NH3 (78×) when the pH was shifted from 6.3 to 8.6 [85]. With this exception in mind, our results support the hypothesis that AOA, AOB, and comammox utilize NH3 as their substrate. As only a few AOM have been characterized in this manner, the characterization of a more diverse set of AOM species is needed in order to make broader conclusions.

It is important to note that the substrate affinities reported here represent whole cell affinities and not the substrate affinity of ammonia monooxygenase (AMO) enzymes. Therefore, further experimental investigation with purified AMO and ammonia/ammonium transporter proteins is warranted. Although NH3 can freely diffuse passively into AOM, this does not mean that the cellular affinity reported here is necessarily unrelated to the transporter-mediated movement of NH3/NH4+ into AOM cells. For example, AOB have previously been shown to accumulate very high (1 M) intracellular NH4+ concentrations [86]. This high intracellular NH4+ concentration may provide a concentrated substrate reservoir, indirectly increasing the concentration of NH3 around the AMO enzyme complex. In addition, the negatively charged S-layer of N. maritimus has been shown to act as a substrate reservoir for the positively charged NH4+. This total ammonium concentration in the pseudo-periplasmic space of AOA, may also act to indirectly increase NH3 concentrations around the AMO enzyme complex [77]. However, the impact of total ammonium capture and transport on the ammonia oxidation kinetic properties of AOB, AOA, and comammox are not yet well understood. It is unknown if such a concentration mechanism would be more important for an AOB with a low substrate affinity (e.g., N. europaea) or for an AOA living in extremely substrate-limited environments (e.g., N. maritimus).

#### The effect of cell morphology

All AOM share the primary enzyme involved in ammonia oxidation, AMO, which is located in the cytoplasmic membrane with its substrate-binding site most likely facing the outside of the cell [77]. Therefore, a higher cellular SA/V ratio likely contributes to an increase in a°, as it increases the space available for AMO and the chance to bind NH3 at very low concentrations. This assumption is based on the hypothesis that an increased abundance of uptake enzymes (e.g., permeases) leads to a higher a° [28, 56]. In fact, the SA/V ratio of AOM (Table S4) correlates to the log of their observed Km(app) for NH3 (R2 = 0.88), Km(app) for total ammonium (R2 = 0.70), a° for NH3 (R2 = 0.78), and a° for total ammonium (R2 = 0.72; Fig. 5). This type of casual semi-log relationship between cell size and nutrient affinities has previously been observed between microorganisms and their affinity for phosphate [87] as well as theoretically modeled for nutrient uptake and utilization in general [88]. Therefore, the SA/V ratio of newly cultured AOM might be a useful general indicator for these cellular kinetic properties. Consequently, AOM with a high SA/V ratio will likely outcompete other AOM in many natural aquatic and terrestrial environments, such as the pelagic marine water column that has a very low standing total ammonium pool. Consistently, these oligotrophic environments have already been postulated to select for organisms with a high SA/V ratio, enhancing their nutrient uptake capabilities [89, 90].

The correlation between the SA/V ratio and cellular kinetic properties of AOM sheds some light on the unusual kinetic properties of the AOA belonging to the genus Nitrosocosmicus. Both ‘Ca. N. oleophilus’ MY3 and ‘Ca. N. franklandus’ C13 possess a very low SA/V ratio compared to other AOA isolates and they both possess several characteristics normally associated with AOB—high substrate tolerances [68,69,70], low affinities for NH3, and a low a° for NH3—that are not consistent with the long-held convention that all AOA are much stronger competitors for NH3 than AOB in substrate-limited environments. Therefore, the individual cell morphology of AOM may have a direct relationship with their cellular kinetic properties. Although this is only a correlation-based observation, it highlights that further investigation into these characteristics is warranted.

In addition to cellular morphology, the size of cell aggregates can affect the kinetic properties of AOM [48]. Cell aggregates have a lower SA/V ratio than individual cells, which can decrease diffusion rates and create microscale substrate/oxygen gradients within aggregates [91]. In order to ensure that the large differences in substrate affinity among AOA are not caused by differences in cell aggregation, the aggregate size of ‘Ca. N. uzonensis’ N4, ‘Ca. N. oleophilus’ MY3, and N. piranensis D3C cultures were inspected before and after MR experiments (Fig. S7).

These three AOA were chosen to highlight the aggregate sizes observed in cultures that displayed the lowest (‘Ca. N. oleophilus’ MY3) and among the highest (‘Ca. N. uzonensis’ N4 and N. piranensis D3C) substrate affinities. No aggregation pattern was observed that would explain the multiple orders of magnitude differences in substrate affinity between these AOA. In fact, of the three AOA investigated, the only strain to form large cell aggregates either before or after MR experiments was N. piranensis D3C, which has one of the highest measured substrate affinities (lowest Km(app) for NH3). In contrast, the cell aggregate size of ‘Ca. N. oleophilus’ MY3 and ‘Ca. N. uzonensis’ N4 were unaffected by the MR experiment and remained relatively small (Fig. S7). As ‘Ca. N. oleophilus’ MY3 has one of the lowest substrate affinities (highest Km(app) for NH3) and formed only small cell aggregates, the low substrate affinity of ‘Ca. N. oleophilus’ MY3 was not an artefact caused by cell aggregation.

Taken together, both environmental (pH and temperature) and AOM cellular (SA/V ratio) factors affect or are related to the observable cellular kinetic properties of individual AOM species. These factors need to be considered when investigating AOM competition or niche differentiation in-situ, as they are often in flux in environmental settings. This can be especially true considering cell morphology, which is often dependent on growth conditions [92]. However, the plasticity of the cellular kinetic properties within individual AOM species does not explain the larger trends observed here across AOA lineages or between AOM (Figs. 2, S1, S35, S8).

## Concluding remarks

In this study we substantially extended the set of available substrate oxidation kinetic properties for AOA by the analysis of pure cultures or enrichments from various lineages within this guild. Furthermore, our kinetic data obtained at different pH values supports the hypothesis that, like for AOB, the substrate for AOA and comammox is NH3. Together, our findings provide novel insights for our understanding of niche differentiation among AOM and demonstrate a surprising variability of the inferred kinetic properties among AOA. Thus, our data strengthens the recent discovery [5] that not all AOA possess an extremely high substrate affinity or specific substrate affinity. The observed links between AOA kinetic properties, phylogeny, and cell morphology also enables the formulation of testable hypotheses on nitrification kinetics in systems thus far characterized solely with molecular (e.g., amplicon sequencing or metagenomic) tools.

As environmental factors such as temperature and pH influence kinetic properties of AOA including their cellular affinity for NH3, future analyses of kinetic properties of AOM should not only be performed at their optimal growth conditions, but also over a range of conditions that reflect their environmental niches. Such experiments will generate a more informative picture on AOM competition and niche differentiation.

### Newly isolated Nitrosotenuis species

The isolated strain N4 is a novel species of the genus Nitrosotenuis of the order Nitrosopumilales, and we propose the following candidate status:

#### Taxonomy

1. (i)

Etymology. The taxonomy for ‘Candidatus Nitrosotenuis uzonensis’ sp. nov. is as follows: Nitrosus (Latin masculine adjective), nitrous; tenuis (Latin masculine adjective), small/slender; uzonensis (Latin neutrum genitive), from Uzon.

2. (ii)

Locality. A terrestrial thermal spring located in the Uzon caldera on the Kamchatka peninsula, Russia.

3. (iii)

Diagnosis. A chemolithoautotrophic ammonia oxidizer of the phylum Thaumarchaeota, which is straight and rod-shaped, with a diameter of 0.2–0.3 µm and a length of 0.4–1.7 µm. Growth over a period of several years has been maintained in a medium with a pH of 7.5 at 37 °C. It belongs to the AOA order Nitrosopumilales (group I.1a). AOA with almost identical 16S rRNA and amoA gene sequences have been detected in various environments, including soil and groundwater [22, 40, 62].


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$There is 2 questions in this pic! Please help ASAP!!$

## This Post Has 7 Comments

1. jumana3 says:

Y=mc squared that’s the formula. yes

2. Giabear23 says:

D, B

Explanation:

William Harvey was the first person to make claims against the Catholic Church in terms of the capabilities of priests to heal the human body.

Newton was someone who showed that the world was something that humans could understand unlike what was accepted in the past about the world being something incomprehensible to humans because God created it. I'm still not quite sure about this one though if you think you  have a better answer then use that one.

3. lildanielmabien says:

Step-by-step explanation: yeah it is cool but you subtract four from the number not subtract your number from four just a heads up

4. Expert says:

step-by-step explanation:

$Which of the following statements are true for the image of a triangles after a dilation that has a$

5. Angel1107 says:

True and Dwarfs are correct

6. 9444 says:

1. Choice (c) is correct. The crust is the outer layer of the earth. Choice (a) is the core of the earth which consists of two parts, the inner and outer core, and is the hottest layer. Choice (b) is the mantle which is solid and rocky.

2. Choice (a) is correct. Image 4 is a plain. Plain is a wide and flat surface of the earth'ss crust where there are few trees standing and is ideal for planting crops, such as rice, corn, wheat, and barley.

3. Choice (d) is correct. Plate tectonics, volcanoes, and continental drift cause the crust's changes. The earth is consists of tectonic plates which slowly moves through time. Because of too much heat from the inmost part of the Earth, the earth's crust suffers from rupture which serves as a passage of magma, volcanic ashes, and gasses. Continental drift is a theory that once upon a time, the Earth has only one continent but with the passage of time, it drifted to where it is today.

4. Choice (b) is correct. Choice (a) happens in the mantle and is moving very slowly because of convection currents. Choice (c) is a movement of plate tectonics in a sideward or downward pattern from the edge of the crust to the mantle. Choice (d) happens after the process of subduction, wherein one plate tectonics may be under the other, and materials above the plates are added.

5. Choice (b) is correct. The layer that separates the crust and outer core is the mantle which is rocky and behaves solidly but as time passes by, it becomes fluid.

6. Choice (d) is correct. Image 1 is a hill. Hills have rounded summit and usually not steep. On the other hand, mountains are land formations that usually have distinct summits, sometimes cone-shaped, and obviously steeper than hills.

format: plain


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Address 3140 34th St N, Saint Petersburg, FL 33713 (727) 528-6600

# lagrange polynomial interpolation error Pinellas Park, Florida

Uniqueness of the interpolating polynomial Proof 1 Suppose we interpolate through n + 1 data points with an at-most n degree polynomial p(x) (we need at least n + 1 datapoints Please try the request again. Your cache administrator is webmaster. Thus, the maximum error will occur at some point in the interval between two successive nodes.

Is there a word for spear-like? The system returned: (22) Invalid argument The remote host or network may be down. Browse other questions tagged numerical-methods interpolation or ask your own question. Privacy policy About Wikipedia Disclaimers Contact Wikipedia Developers Cookie statement Mobile view ERROR The requested URL could not be retrieved The following error was encountered while trying to retrieve the URL:

In the case of Karatsuba multiplication this technique is substantially faster than quadratic multiplication, even for modest-sized inputs. share|cite|improve this answer answered Feb 11 at 13:38 lorena 11 This question already had a well-accepted answer. Why aren't there direct flights connecting Honolulu, Hawaii and London, UK? The situation is rather bad for equidistant nodes, in that uniform convergence is not even guaranteed for infinitely differentiable functions.

Wikipedia® is a registered trademark of the Wikimedia Foundation, Inc., a non-profit organization. Please try the request again. Either way this means that no matter what method we use to do our interpolation: direct, Lagrange etc., (assuming we can do all our calculations perfectly) we will always get the Appunti di Calcolo Numerico.

For example, given a = f(x) = a0x0 + a1x1 + ... Is there a way to view total rocket mass in KSP? Your cache administrator is webmaster. This turns out to be equivalent to a system of simultaneous polynomial congruences, and may be solved by means of the Chinese remainder theorem for polynomials.

So we can get Y ( n + 1 ) ( t ) = R n ( n + 1 ) ( t ) − R n ( x ) W Can I stop this homebrewed Lucky Coin ability from being exploited? Since $f''$ is strictly increasing on the interval $(1, 1.25)$, the maximum error of ${f^{2}(\xi(x)) \over (2)!}$ will be $4e^{2 \times 1.25}/2!$. Belg. (in French), 4: 1–104 Brutman, L. (1997), "Lebesgue functions for polynomial interpolation — a survey", Ann.

Your cache administrator is webmaster. Is it correct to write "teoremo X statas, ke" in the sense of "theorem X states that"? BIT. 33 (33): 473–484. more stack exchange communities company blog Stack Exchange Inbox Reputation and Badges sign up log in tour help Tour Start here for a quick overview of the site Help Center Detailed

Here, the interpolant is not a polynomial but a spline: a chain of several polynomials of a lower degree. For better Chebyshev nodes, however, such an example is much harder to find due to the following result: Theorem. Alistair (1980), Approximation Theory and Numerical Methods, John Wiley, ISBN0-471-27706-1 External links Hazewinkel, Michiel, ed. (2001), "Interpolation process", Encyclopedia of Mathematics, Springer, ISBN978-1-55608-010-4 ALGLIB has an implementations in C++ / C# But r(x) is a polynomial of degree ≤ n.

numerical-methods interpolation share|cite|improve this question edited Feb 16 '15 at 20:34 asked Feb 16 '15 at 20:01 Alex 614 add a comment| 2 Answers 2 active oldest votes up vote 2 And we know that there has to exist a critical point between each of the zeros so comparing the norms of each of the critical points always gives us the max Another method is to use the Lagrange form of the interpolation polynomial. Furthermore, you only need to do O(n) extra work if an extra point is added to the data set, while for the other methods, you have to redo the whole computation.

GSL has a polynomial interpolation code in C Interpolating Polynomial by Stephen Wolfram, the Wolfram Demonstrations Project. Interpolation error This section may be confusing or unclear to readers. (June 2011) (Learn how and when to remove this template message) When interpolating a given function f by a polynomial Interpolation based on those points will yield the terms of W(x) and subsequently the product ab. Suppose that the interpolation polynomial is in the form p ( x ) = a n x n + a n − 1 x n − 1 + ⋯ + a

Plugging in $x=1.4$ in the formula above gives us $1.461899$. How do spaceship-mounted railguns not destroy the ships firing them? The resulting formula immediately shows that the interpolation polynomial exists under the conditions stated in the above theorem. Hermite interpolation problems are those where not only the values of the polynomial p at the nodes are given, but also all derivatives up to a given order.

The technique of rational function modeling is a generalization that considers ratios of polynomial functions. Does there exist a single table of nodes for which the sequence of interpolating polynomials converge to any continuous function f(x)? For equally spaced intervals In the case of equally spaced interpolation nodes where x 0 = a {\displaystyle x_{0}=a} and x i = a + i h {\displaystyle x_{i}=a+ih} , for Proof Set the error term as R n ( x ) = f ( x ) − p n ( x ) {\displaystyle R_{n}(x)=f(x)-p_{n}(x)} and set up an auxiliary function: Y

This suggests that we look for a set of interpolation nodes that makes L small. Generated Tue, 18 Oct 2016 16:16:51 GMT by s_ac4 (squid/3.5.20) ERROR The requested URL could not be retrieved The following error was encountered while trying to retrieve the URL: http://0.0.0.7/ Connection Specifically, we know that such polynomials should intersect f(x) at least n + 1 times. Jahr. (in German), 23: 192–210 Powell, M.

and b = g(x) = b0x0 + b1x1 + ..., the product ab is equivalent to W(x) = f(x)g(x). In this case, we can reduce complexity to O(n2).[5] The Bernstein form was used in a constructive proof of the Weierstrass approximation theorem by Bernstein and has nowadays gained great importance


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# 15 Different Outputs and Extensions

There are many different outputs for rmarkdown - as a start, THIS VERY BOOK IS WRITTEN IN RMARKDOWN! How cool is that?

## 15.1 Alternative output formats

### 15.1.2 For Journals

Note that if you are using a template and want the full featured citation features,

your YAML will need to include something like this:

output:
bookdown::pdf_book:
base_format: rticles::plos_article


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# every algebraically closed field is perfect

###### Proposition 1.

Every algebraically closed field is perfect

###### Proof.

Let $K$ be an algebraically closed field of prime characteristic $p$. Take $a\in K$. Then the polynomial $X^{p}-a$ admits a zero in $K$. It follows that $a$ admits a $p$th root in $K$. Since $a$ is arbitrary we have proved that the field $K$ is perfect.∎

Title every algebraically closed field is perfect EveryAlgebraicallyClosedFieldIsPerfect 2013-03-22 16:53:06 2013-03-22 16:53:06 polarbear (3475) polarbear (3475) 6 polarbear (3475) Result msc 12F05


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EPS-HEP2021 conference

26-30 July 2021
Zoom
Europe/Berlin timezone

Comparison of $pp$ and $p \bar{p}$ differential elastic cross sections and observation of the exchange of a colorless $C$-odd gluonic compound

26 Jul 2021, 11:45
15m
Zoom

Zoom

Parallel session talk QCD and Hadronic Physics

Speaker

Christophe Royon (The University of Kansas (US))

Description

We describe an analysis comparing the $p\bar{p}$ elastic cross section as measured by the D0 Collaboration at a center-of-mass energy of 1.96~TeV to that in $pp$ collisions as measured by the TOTEM Collaboration at 2.76, 7, 8, and 13 TeV using a model-independent approach. The TOTEM cross sections, extrapolated to a center-of-mass energy of $\sqrt{s} =$ 1.96 TeV, are compared with the D0 measurement in the region of the diffractive minimum and the second maximum of the $pp$ cross section.
The two data sets disagree at the 3.4$\sigma$ level and thus provide evidence for the $t$-channel exchange of a colorless, $C$-odd gluonic compound, also known as the odderon.
We combine these results with a TOTEM analysis of the same $C$-odd exchange based on the total cross section and the ratio of the real to imaginary parts of the forward elastic strong interaction scattering amplitude in $pp$ scattering.
The combined significance is larger than 5$\sigma$ and is interpreted as the first observation of the exchange of a colorless, $C$-odd gluonic compound.

Collaboration / Activity D0 / TOTEM Christophe Royon christophe.royon@cern.ch

Primary author

Christophe Royon (The University of Kansas (US))


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# Present Value of $4 in 96 Years When you have a single payment that will be made to you, in this case$4, and you know that it will be paid in a certain number of years, in this case 96 years, you can use the present value formula to calculate what that $4 is worth today. Below is the present value formula we'll use to calculate the present value of$4 in 96 years.

$$Present\: Value = \dfrac{FV}{(1 + r)^{n}}$$

We already have two of the three required variables to calculate this:

• Future Value (FV): This is the $4 • n: This is the number of periods, which is 96 years So what we need to know now is r, which is the discount rate (or rate of return) to apply. It's worth noting that there is no correct discount rate to use here. It's a very personal number than can vary depending on the risk of your investments. For example, if you invest in the market and you earn on average 8% per year, you can use that number for the discount rate. You can also use a lower discount rate, based on the US Treasury ten year rate, or some average of the two. The table below shows the present value (PV) of$4 paid in 96 years for interest rates from 2% to 30%.

As you will see, the present value of $4 paid in 96 years can range from$0.00 to $0.60. Discount Rate Future Value Present Value 2%$4 $0.60 3%$4 $0.23 4%$4 $0.09 5%$4 $0.04 6%$4 $0.01 7%$4 $0.01 8%$4 $0.00 9%$4 $0.00 10%$4 $0.00 11%$4 $0.00 12%$4 $0.00 13%$4 $0.00 14%$4 $0.00 15%$4 $0.00 16%$4 $0.00 17%$4 $0.00 18%$4 $0.00 19%$4 $0.00 20%$4 $0.00 21%$4 $0.00 22%$4 $0.00 23%$4 $0.00 24%$4 $0.00 25%$4 $0.00 26%$4 $0.00 27%$4 $0.00 28%$4 $0.00 29%$4 $0.00 30%$4 $0.00 As mentioned above, the discount rate is highly subjective and will have a big impact on the actual present value of$4. A 2% discount rate gives a present value of $0.60 while a 30% discount rate would mean a$0.00 present value.

The rate you choose should be somewhat equivalent to the expected rate of return you'd get if you invested $4 over the next 96 years. Since this is hard to calculate, especially over longer periods of time, it is often useful to look at a range of present values (from 5% discount rate to 10% discount rate, for example) when making decisions. Hopefully this article has helped you to understand how to make present value calculations yourself. You can also use our quick present value calculator for specific numbers. ### Link To or Reference This Page If you found this content useful in your research, please do us a great favor and use the tool below to make sure you properly reference us wherever you use it. We really appreciate your support! • "Present Value of$4 in 96 Years". StudyFinance.com. Accessed on June 24, 2021. https://studyfinance.com/present-value/4-in-96-years/.

• "Present Value of $4 in 96 Years". StudyFinance.com, https://studyfinance.com/present-value/4-in-96-years/. Accessed 24 June, 2021 • Present Value of$4 in 96 Years. StudyFinance.com. Retrieved from https://studyfinance.com/present-value/4-in-96-years/.


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# Quantify Drop-off for digital PCR assays with Crystal Miner™

## How to interpret Drop-Off digital PCR assays using Crystal Miner

• A 1st channel to detect the Drop-Off Probe that spans the mutation site but is uniquely complementary to the wild-type sequence (e.g. “blue Detection Channel” on the vertical axis in Fig. 1)
• A 2nd channel to detect the Reference Probe that hybridizes adjacent to the mutation site and is complementary to both the mutant and wild-type alleles (e.g. “Green Detection Channel” on the horizontal axis in Fig. 1)

In the presence of a wild-type allele, both the drop-off and reference probes willl hybridize with their target, leading to a double positive signal (turquoise population in Fig. 1). Whereas in the presence of a mutant allele, even a single nucleotide mutation is enough to destabilize the hybridization of the drop-off probe so that only the reference probe anneals to its target leading to a single positive signal (green population in Fig. 1)

For more information on Drop-off assay design, see our Application Note “Drop-off Crystal Digital PCR™ detects NRAS, KRAS, EGFR mutations” at https://www.stillatechnologies.com/application-notes/

Figure 1: A) Crystal Miner 2D dot-plot of a drop-off assay showing three clusters: double-positive droplets (turquoise) containing wild-type alleles, single positive droplets (green) containing mutant alleles, and double negative droplets (black). B) During droplet generation, a fraction of mutant and wild-type alleles can be randomly co-encapsulated within the same droplet (turquoise). Although the proportion of non-double-positive droplets (green or black) containing mutant alleles (green). M , mutation.

## Step-by-step method to quantify Drop-Offs using Crystal Miner

The wild concentration $C_{WT}$ can be directly derived from the proportion $P_{WT}$ of droplets that are double-positive (whether they are positive or negative in a 3rd detection channel). However, the drop-off mutant concentration $C_{MUT}$ must be derived from the proportion $P_{MUT}$ of droplets that confidently contains mutant alleles. Due to the potential for wild-type and mutant allele co-encapsulation, double-positive droplets are ambiguous droplet for mutant quantification. As such, $P_{MUT}$ can only be determined by excluding the double-positive droplets and thus restricting the mutant positive population to the droplets that are positive for the Reference Probe but negative for the Drop-Off Probe (whether they are positive or negative in 3rd detection channel).

To quantify Drop-off populations using Crystal Miner, follow the steps below:

1. Define the three droplet populations: mutant positive, mutant negative, wild-type positive

• Go to ANALYZE DATA > Plots & Populations > 2D dot plot
• Select the “Polygon” type for the thresholding mode, then select your well(s) of interest in the left-hand panel
• In the 2D dot plot showing the “Drop-Off Detection Channel” on the vertical axis and the “Reference Detection Channel” on the horizontal axis (double-click on the axes to modify their Channel attributions if needed), draw a polygon around each of the following three droplet populations of interest:
• Mutant positive population, discounting double positives (e.g. “EGFR del 19 Mut” in Fig. 2)
• Mutant negative population, discounting double positives (e.g. “EGFR del 19 Mut – Negatives” in Fig. 2)
• Wild-type positive population, i.e. the double positive (e.g. “EGFR del 19 WT” in Fig. 2)

To draw a polygon: press and hold the Ctrl key and right-click to create each vertex, then release the Ctrl key before the last right-click to close the polygon. You can then name the droplet population contained in the polygon and choose its color.

2. Specify the negative population associated with the mutant positive population/p>

• Go to ANALYZE DATA > Plots & Populations > Population Editor
• In the “Population” window on the right: select the mutant positive population, then click on “EDIT”, expand the option “Need to define specific negative droplets” and click on “Edit Negatives”
• In the “Zones List” window in the middle: check the box corresponding to the mutant negative population in the “Negative” column (e.g. the zone “2” in Fig 2)
• In the “Population” window on the right: click on “APPLY”

Figure 2: Example of Crystal Miner’s display for a digital PCR experiment targeting the drop-off mutant EGFR Del19 using two channels (channel 1: reference probe with YY fluorophore; channel 2: drop-off probe with FAM fuorophore). In the case where the 3rd detection channel is used, for example to detect an unrelated mutation or an internal control, the polygon zones will project in the 3rd dimension so they may contain droplets that are negative or positive for the 3rd detection channel. The use of a 3rd channel will in no way impact reliable drop-off quantification.

3. Calculate the MAF of a drop-off mutant

• Go to VIEW RESULTS > Result Table and get the drop-off mutant concentration $C_{MUT}$ and the wild-type concentration $C_{WT}$ automatically estimated by Crystal Miner
• Go to Export and export all result data
• Deduce the MAF of drop-off mutant by applying the following formula: $MAF_{dropoff}$ = $C_{MUT}$ ( $C_{WT}$ + $C_{MUT}$ )

Notes

The relative loss of Limit of Detection (LOD) for a drop-off mutant compared to the LOD of a point mutant increases exponentially with the expected wild-type concentration (e.g. LOD at 95% confidence level raises from 0.2 to 0.27 cp/µL for a wild-type concentration $C_{WT}$ =500 cp/µL in the loaded sample of one Sapphire chip well).

If the double-positive droplets are not excluded and instead counted as mutant negative droplets, the MAF of the drop-off mutant will be underestimated almost linearly according to the expected wild-type concentration $C_{WT}$ and with a nearly constant factor for any $C_{MUT}$ $\leq$ 100 cp/µL (e.g. underestimation around 25% for $C_{WT}$ =500 cp/µL in the loaded sample). ln addition, keeping the double-positive droplets would neither decrease the relative uncertainty of the measured drop-off mutant concentration nor the LOD of the drop-off mutant.

## Technical Note Highlights

• By using the NaicaCrystal Miner advanced Population Editor feature, ambiguous droplets can be discounted ensuring accurate quantification of drop-off mutations
• Excluding ambiguous droplets avoids underestimating the concentration of the drop-off mutant without decreasing the concentration measurement precision

• Because only two detection channels are required to estimate a drop-off mutant allelic fraction, the third detection channel can be used, for example, to add an internal amplification control or to simultaneously quantify the MAF of a point mutation (see the Application Note “Drop-off Crystal Digital PCR detects NRAS, KRAS & EGFR mutations” at https://www.stillatechnologies.com/application-notes/).


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# ML: Clustering¶

Clustering is one of the types of unsupervised learning. It is similar to classification: the aim is to give a label to each data point. However, unlike in classification, we are not given any examples of labels associated with the data points. We must infer from the data, which data points belong to the same cluster. This can be achieved using some notion of distance between the data points. Data points in the same cluster are somehow close to each other.

One of the simplest clustering methods is the k-means clustering. It aims at producing a clustering that is optimal in the following sense:

• the centre of each cluster is the average of all points in the cluster
• any point in a cluster is closer to its centre than to a centre of any other cluster

The k-means clustering is first given the wanted number of clusters, say k, as a hyperparameter. Next, to start the algorithm, k points from the data set are chosen randomly as cluster centres. Then the following phases are repeated iteratively:

• any data point is set to belong to a cluster, whose centre is closest to it
• then for each cluster a new centre is chosen as the average of the data points in the cluster

This procedure is repeated until the clusters no longer change. This kind of algorithm is called an Expectation-Maximization (EM) algorithm, which is known to converge.

## Simple example¶

The scikit-learn library has an implementation of the k-means algorithm. Let’s apply it to a set of randomly generated blobs, whose labels we throw away.

[1]:

import numpy as np
import matplotlib.pyplot as plt
%matplotlib inline

[2]:

from sklearn.datasets import make_blobs
X,y = make_blobs(centers=4, n_samples=200, random_state=0, cluster_std=0.7)
print(X[:10],y[:10])

[[ 2.26403424  1.82613379]
[-0.97647444  2.59138889]
[ 1.10046838  4.02254067]
[-2.82715074  7.11970523]
[ 1.53393915  0.31915668]
[ 0.98362009  5.55389667]
[-1.74452433  2.98606238]
[ 0.35482006  2.9172298 ]
[ 1.83747356  5.14545322]
[ 1.48663347  4.39407536]] [1 2 0 3 1 0 2 0 0 0]

Now we plot these points, but without coloring the points using the labels:

[3]:

plt.scatter(X[:,0],X[:,1]);

We can still discern four clusters in the data set. Let’s see if the k-means algorithm can recover these clusters. First we create the instance of the k-means model by giving it the number of clusters 4 as a hyperparameter.

[4]:

from sklearn.cluster import KMeans
model = KMeans(4)
model.fit(X)
print(model.cluster_centers_)

[[-1.36512238  7.70188306]
[ 2.07464749  0.9869902 ]
[-1.70639178  2.9104771 ]
[ 0.86008475  4.31501411]]

[5]:

plt.scatter(X[:,0],X[:,1], c=model.labels_);
plt.scatter(model.cluster_centers_[:,0], model.cluster_centers_[:,1], s=100, color="red"); # Show the centres

The clustering looks more or less correct. To get a more quantitative measure of success we can get the accuracy score.

[6]:

from sklearn.metrics import accuracy_score
acc=accuracy_score(y, model.labels_)
print("Accuracy score is", acc)

Accuracy score is 0.49

Oops! Even though the clusters could match almost perfectly to the original, their labels might be permuted. Let’s select randomly one point from each cluster and check their labels from the original data labels. Then we use this label for the whole cluster. In essence, we are renaming the clusters, not re-clustering the data.

[7]:

import scipy
def find_permutation(n_clusters, real_labels, labels):
permutation=[]
for i in range(n_clusters):
idx = labels == i
new_label=scipy.stats.mode(real_labels[idx])[0][0]  # Choose the most common label among data points in the cluster
permutation.append(new_label)
return permutation

[8]:

permutation = find_permutation(4, y, model.labels_)
print(permutation)

[3, 1, 2, 0]

[9]:

new_labels = [ permutation[label] for label in model.labels_]   # permute the labels
print("Accuracy score is", accuracy_score(y, new_labels))

Accuracy score is 0.99

So, the k-means algorithm seems to work well in this case, but there can be several problems. Firstly, even though an EM algorithm always converges, it might converge to a local maximum. To avoid this, EM type algorithms are usually run several times, each time starting from different random initial values. For instance, in the scikit-learn implementation, the algorithms is restarted by default 10 times.

## More complicated example¶

The k-means algorithm can have difficulties when the clusters are not convex shapes:

[10]:

from sklearn.datasets import make_moons
X,y = make_moons(200, noise=0.05, random_state=0)

[11]:

plt.scatter(X[:,0], X[:,1]);

[12]:

model=KMeans(2)
model.fit(X)
plt.scatter(X[:,0], X[:,1], c=model.labels_);

The clustering does not work well now, since it is not possible to separate the two clusters with a line. We could embed this data set into a higher dimensional space, where the separation is possible. And then apply the k-means clustering.

Alternatively, we can use a different type of clustering algorithm for this case. The DBSCAN algorithm is based on densities and works well on data whose density in the clusters is uniform.

[13]:

from sklearn.cluster import DBSCAN
model = DBSCAN(eps=0.3)
model.fit(X)
plt.scatter(X[:,0], X[:,1], c=model.labels_);

The good news is that DBSCAN does not require the user to specify the number of clusters. But now the algorithm depends on another hyperparameter: a threshold for distance (here 0.3).

## Clustering digits¶

Using scikit-learn we can download a set of 1797 images of handwritten digits with the correct labels 0,1,…,9. The images have quite a low resolution: 8*8=64 pixels. Let’s see how our machine learning method works with this kind of data.

[14]:

from sklearn.datasets import load_digits
digits.data.shape

[14]:

(1797, 64)

To get an idea how these data points look like, we plot first ten of these.

[15]:

fig, axes = plt.subplots(2,5, subplot_kw=dict(xticks=[], yticks=[]))
for ax, digit in zip(axes.flat, digits.data[:10]):
ax.imshow(digit.reshape(8,8), cmap="gray")

Let’s cluster these data points into ten clusters.

[16]:

model=KMeans(n_clusters = 10, random_state=0)
model.fit(digits.data)
model.cluster_centers_.shape

[16]:

(10, 64)

So, we have ten cluster centres, which are images with 8x8=64 pixels in them. We can have a look at their appearence:

[17]:

fig, axes = plt.subplots(2,5, subplot_kw=dict(xticks=[], yticks=[]))
for ax, digit in zip(axes.flat, model.cluster_centers_):
ax.imshow(digit.reshape(8,8), cmap="gray")

One can recognize these numbers with the exception of maybe number eight. What is the accuracy score of this clustering?

[18]:

permutation3 = find_permutation(10, digits.target, model.labels_)
print(permutation3)
acc = accuracy_score(digits.target, [ permutation3[label] for label in model.labels_])
print("Accuracy score is", acc)

[4, 3, 5, 9, 7, 0, 1, 8, 2, 6]
Accuracy score is 0.793544796884

This is quite a good result for such a simple algorithm!

Using the same iris data set that you saw earlier in the classification, apply k-means clustering with 3 clusters. Create a function plant_clustering that loads the iris data set, clusters the data and returns the accuracy_score.

This exercise can give four points at maximum!

Read the tab separated file data.tsv from the src folder into a DataFrame. The dataset has two features X1 and X2, and the label y. Cluster the feature matrix using DBSCAN with different values for the eps parameter. Use values in np.arange(0.05, 0.2, 0.05) for clustering. For each clustering, collect the accuracy score, the number of clusters, and the number of outliers. Return these values in a DataFrame, where columns and column names are as in the below example.

Note that DBSCAN uses label -1 to denote outliers , that is, those data points that didn’t fit well in any cluster. You have to modify the find_permutation function to handle this: ignore the outlier data points from the accuracy score computation. In addition, if the number of clusters is not the same as the number of labels in the original data, set the accuracy score to NaN.

     eps   Score  Clusters  Outliers
0    0.05      ?         ?         ?
1    0.10      ?         ?         ?
2    0.15      ?         ?         ?
3    0.20      ?         ?         ?

Before submitting the solution, you can plot the data set (with clusters colored) to see what kind of data we are dealing with.

Points are given for each correct column in the result DataFrame.

## Hierarchical clustering¶

Hierarchical clustering works by first putting each data point in their own cluster and then merging clusters based on some rule, until there are only the wanted number of clusters remaining. For this to work, there needs to be a distance measure between the data points. With this distance measure d, we can define another distance measure between the clusters U and V using one of the following methods (linkages):

• single: $$d(U, V) := \min_{u \in U, v \in V} d(u,v)$$
• complete: $$d(U, V) := \max_{u \in U, v \in V} d(u,v)$$
• average: $$d(U, V) := \sum_{u \in U, v \in V} \frac{d(u,v)}{|U||V|}$$
• ward: tries to minimize the variance in each cluster

At each iteration of the algorithm two clusters that are closest to each other are merged. After this the distance between the clusters are recomputed, and then it continues to the next iteration.

Below is an example with a botanical dataset with 150 samples from three species. Each species appears in the dataset 50 times. Each sample point has 4 features, which are basically dimensions of the “leaves” of the flower.

We use the seaborn library to both to compute the clustering and to visualize the result. The visualization consists of two parts: the heatmap, whose rows and/or columns may be reordered so as to have the elements of the same cluster next to each other; and the dendrogram, which shows the way the clusters were merged. The colors give the length of the corresponding features.

[19]:

import seaborn as sns; sns.set(color_codes=True)
species = iris.pop("species")   # Remove the species column
print(species.unique())         # The samples seems to be from these three species
sns.clustermap(iris, method="ward", col_cluster=False, cbar_kws={'label': 'centimeters'}); # Cluster only the rows
#plt.colorbar().ax.set_title('This is a title')
#plt.gca().images[-1].colorbar.ax.set_title("title")

['setosa' 'versicolor' 'virginica']

With sharp eye and good will one can discern three clusters in the above heatmap and dendrogram.

This exercise can give three points at maximum!

A binding site is a piece of DNA where a certain protein prefers to bind. The piece of DNA can be described as a string consisting of letters A, C, G, and T, which correspond to nucleotides Adenine, Cytosine, Guanine, and Thymine. In this exercise the length of binding sites is eight nucleotides. They are stored in the file data.seq, and the binding sites there are classified into two classes.

Part 1. Write function toint that converts a nucleotide to an integer. Use the following mapping:

A -> 0
C -> 1
G -> 2
T -> 3

Write also function get_features_and_labels that gets a filename as a parameter. The function should load the contents of the file into a DataFrame. The column X contains a string. Convert this column into a feature matrix using the above toint function. For example the column ["GGATAATA","CGATAACC"] should result to the feature matrix

[[2,2,0,3,0,0,3,0],
[1,2,0,3,0,0,1,1]]

The function should return a pair, whose first element is the feature matrix and the second element is the label vector.

Part 2. Create function cluster_euclidean that gets a filename as parameter. Get the features and labels using the function from part 1. Perform hierarchical clustering using the function sklearn.cluster.AgglomerativeClustering. Get two clusters using average linkage and euclidean affinity. Fit the model and predict the labels. Note that you may have to use the find_permutation function again, because even though the clusters are correct, they may be labeled differently than the real labels given in data.seq. The function should return the accuracy score.

Part 3. Create function cluster_hamming that works like the function in part 2, except now using the hamming affinity. Even though it is possible to pass the function hamming to AgglomerativeClustering, let us now compute the Hamming distance matrix explicitly. We can achieve this using the function sklearn.metrics.pairwise_distances. Use the affinity parameter precomputed to AgglomerativeClustering. And give the distance matrix you got from pairwise_distances, instead of the feature matrix, to the fit_predict method of the model. If you want, you can visualize the clustering using the provided plot function.

Which affinity (or distance) do you think is theoretically more correct of these two (Euclidean or Hamming)? Why?


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# 청색(靑色) 사파이어의 열처리(熱處理)와 분광학적(分光學的) 특성(特性)

• Na, Kyung-Ju (Department of Geology, Chungnam National University) ;
• Kim, Won-Sa (Department of Geology, Chungnam National University) ;
• Kim, Mun-Young (Korea Institute of Geology, Mining and Materials) ;
• Bae, In-Kook (Korea Institute of Geology, Mining and Materials) ;
• Jang, Young-Nam (Korea Institute of Geology, Mining and Materials)
For the blue sapphires from Santung, China, the color change before and after has been investigated by UV-Visible spectrophotometry method. The blue sapphires from Shantung show four groups of absorption bands: the bands A (374, 386 and 450 nm) being attributed to single $Fe^{3+}$ ion, the band B (560, 579 and $704n{\breve{m}}$) to $Fe^{2+}$/$Ti^{4+}$ pairs, the band C (-800 nm) to $Fe^{2+}$/$Fe^{3+}$ pairs, and the D (528 nm) to $Ti^{3+}$ dd transitions. From those UV-VIS characteristics the origin of blue color of the sapphires is confirmed to be attributed by the factors such as $Fe^{2+}$/$Fe^{3+}$ and $Ti^{3+}$/$Ti^{4+}$. The absorption spectra of natural blue sapphires before and after heat treatment show distintive features, comparing with those of sapphires from other localities: the bands of 689 nm and of $Cr^{3+}$ are not recorded on the spectra of sapphires from Shantung. The band (492 nm), which resulted from $Ti^{3+}$, is not shown and the intensity of the band 528 nm decreases after the heat treatment. Decoloration of ink-blue sapphires are found to be successful by heat treatment with the control of annealing and atmosphere. During the diffusion process the excess components of impurities contained originally in the host crystal were expelled to the surface of crystals, enhancing the transparency of the crystals noticeably.


text:
COMPSTAT 2016: Start Registration
View Submission - COMPSTAT
A0503
Title: yuimaGUI: A graphical user interface for computational finance based on the yuima R package Authors:  Emanuele Guidotti - University of Milan (Italy) [presenting]
Stefano Iacus - University of Milan (Italy)
Lorenzo Mercuri - University of Milan (Italy)
Abstract: The aim of Yuima project is to develop a complete environment for simulation and inference of Stochastic Differential Equations (SDE) via an R package called yuima. The package is developed using the object oriented programming language S4 and allows the user to manage a stochastic process characterized by a SDE with the following general form: $dXt =b(t,Xt,\theta)dt+a(t,Xt,\theta)dWHt +c(t,Xt,\theta)dZt$, where $b(t,Xt,\theta)$, $a(t,Xt,\theta)$ and $c(t,Xt,\theta)$ are functions defined by the user. $WH$ is the fractional Brownian motion where the Hurst coefficient $H$ is fixed by default to 1/2 that corresponds to the standard Brownian motion.The yuima package has also estimation and simulation routines for two models widely used in modern computational finance: the Continuous ARMA$(p,q)$ driven by a general Levy process and the COGARCH$(p,q)$ model for high frequency data. Moreover, yuimaGUI allows to simulate a portfolio of assets and derivatives under different scenarios and further evaluate thee overall returns along with their distribution. After a brief presentation of the yuima framework, we will focus on aspects related to computational finance and show how they can be easily approached through the yuimaGUI package. We will go through data mining and clustering of financial time series, model selection, change point estimation, scenario simulation and the analysis of the distribution of the expected returns for composite portfolios.


text:
# Simple automorphisms of finite relations

Finding automorphisms is a hard problem in general, but I am studying some simple subgroups of automorphisms, which are easy to find.

I have some r-ary relation R on a finite set U (if it was a binary relation, it would be a graph, but I am considering any general relation).

Let A2 be automorphisms $(a/b)(c/d)$ consisting of exactly two disjoint transpositions, such that none of these transposition is an automorphism by itself. Let's call one transposition to be a witness of another one. There is a relation L2 on U, such that $L2(a,b)$ iff the transposition (a/b) is a part of some automorphism in A2.

I need to represent the subgroup $G$ of automorphisms generated by A2. I am making an algorithm, which gets a relation R, then gets a permutation P and has to decide, if $P \in G$.

We draw a transposition as a two-sided black arrow between two circles. A witness is a yellow line between two transpositions. I have proven two theorems:

1. L2 is transitive: Let L2(u1,u2), L2(u2,u3). Then L2(u1,u3). There are three ways such automorphisms can interact (with 1, 2 or 3 common values). Composing such automorphisms multiple times gives us new automorphisms of A2 containing (u1/u3).
2. If $(a/b)(c/d), (c/d)(e/f)$ are automorphisms, then $(c/d)$ is an automorphism.

The second theorem actually tells us, that each transposition can have at most one witness (otherwise a single transposition would be an automorphism, which violates the definition of A2). This means, that automorphisms of A2 can not interact as shown in a) - (u1/u3) is a second witness of (a/b), neither c) - (v1/v2) has two witnesses. They can interact only as shown in b).

L2 splits U into equivalence classes. These classes group into pairs of the same size, such that every two elements of one class have some witness transposition in another class.

To check, if $P \in G$, I first check, if P keeps each element inside its L2-equivalency class.

I thought, that composing elements of A2 creates "ladders", that there is 1-to-1 mapping F between two L2-equivalence classes ($F(u1)=v1, F(u2)=v2, F(u3)=v3$) and it will be enough to check, that $P(a)=b \implies P(F(a)) = F(b)$, i.e. that automorphisms in $G$ can shuffle the elements in any way possible, as long as each element remains in its L2-equivalency class and the same shuffling is performed on the other side of the ladder.

But what if there is $(u0/u1)(v0/v2) \in A2$, which will break the ladder structure, and then the map F makes no sense? I can not disprove it. Can you see any other useful properties, which can help me decide, if $P \in G$?

Example: $R = \{(0,2,3), (1,3,2)\}$, the automorphism is $(0/1)(2/3)$, however, the map F is not unique (you can match 0 with 2, 1 with 3, or 0 with 3 and 1 with 2). But I think, that when L2-classes are larger than 2 (taller ladder), F will be unique.

• What do you mean by the notation $(a/b)$? – Pat Devlin Dec 5 '16 at 5:26
• $(a/b)$ is a transposition of a and b. – IvanKuckir Dec 5 '16 at 9:06


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### Month: August 2022

August 23, 2022 Systems No comments

Google Colaboratory is a popular service for running Jupyter notebooks in a cloud environment using the computational resources provided by Google.

As with other cloud services, it is possible to configure Google Colab to work with Madagascar. The solution is shown in this notebook.

We propose a method to efficiently measure time shifts and scaling functions between seismic images using amplitude-adjusted plane-wave destruction filters. Plane-wave destruction can efficiently measure shifts of less than a few samples, making this algorithm particularly effective for detecting small shifts. Separating shifts and scales allows shifting functions to be measured more accurately. When shifts are large, amplitude-adjusted plane-wave destruction can also be used to refine shift estimates obtained by other methods. The effectiveness of this algorithm in predicting shifting and scaling functions is demonstrated by applying it to a synthetic trace and a time-lapse field data example from the Cranfield CO$_2$ sequestration project.