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# Antagonistic odor interactions in olfactory sensory neurons are widespread in freely breathing mice

## Abstract

Odor landscapes contain complex blends of molecules that each activate unique, overlapping populations of olfactory sensory neurons (OSNs). Despite the presence of hundreds of OSN subtypes in many animals, the overlapping nature of odor inputs may lead to saturation of neural responses at the early stages of stimulus encoding. Information loss due to saturation could be mitigated by normalizing mechanisms such as antagonism at the level of receptor-ligand interactions, whose existence and prevalence remains uncertain. By imaging OSN axon terminals in olfactory bulb glomeruli as well as OSN cell bodies within the olfactory epithelium in freely breathing mice, we find widespread antagonistic interactions in binary odor mixtures. In complex mixtures of up to 12 odorants, antagonistic interactions are stronger and more prevalent with increasing mixture complexity. Therefore, antagonism is a common feature of odor mixture encoding in OSNs and helps in normalizing activity to reduce saturation and increase information transfer.

## Introduction

The number of distinct types of sensory neurons is usually far smaller than the number of distinct stimuli that an animal needs to detect, necessitating individual neurons to be receptive to multiple stimuli. This feature is especially relevant in the olfactory system where the number of odorous molecules vastly exceeds the repertoire of receptor types1,2,3. Further, the olfactory world at any instant consists of complex mixtures of odorants, with individual olfactory receptors encountering multiple odorants at once (Fig. 1a). Therefore, how multiple ligands interact at single odorant receptors will define the mode of coding in, and the capacity of, the olfactory system. A large fraction of the olfactory sensory neurons (OSNs) in mammals express a family of odorant receptors that are G-protein coupled receptors4,5,6. If the ligand-receptor binding kinetics are modeled with a single affinity parameter, simultaneous activation by multiple ligands will be simply determined by the relative affinities, until saturation7,8,9,10. However, it is becoming increasingly clear that more complex interactions can occur.

Recent theoretical work has shown that a variety of nonlinear interactions among multiple ligands at the same receptor can readily arise with a simple two-step model of receptor activation11 (Fig. 1b, c). Experimental work in vitro in OSNs has suggested the existence of nonlinear interactions, especially antagonism or partial antagonism12,13,14,15,16,17,18,19,20. However, the prevalence of these interactions, especially in living animals within the constraints of natural sniffing dynamics, has not been explored. At a more basic level, evidence for multistep receptor activation has also been sparse. For example, odorants with different affinities for a given receptor could also have distinct maximal activation if their efficacies are different17, but few studies have systematically explored this aspect.

Elucidating the principles of mixture interactions in OSNs is important for understanding odor coding. Odor identity and abundance are widely accepted to be represented by a combination of OSNs21,22,23. While each odorant activates a discrete pattern of sensory inputs, there can be considerable overlap in patterns of OSN activation corresponding to different odorants24,25,26,27. Since naturalistic odor stimuli are complex blends of many odorants, even relatively simple odor blends may saturate a large fraction of the complement of OSNs, thereby limiting their information coding capacity. More generally, even before saturation sets in, nonlinear interactions among multiple odorants in a given OSN may pose challenges for downstream decoding of odor identity28,29.

We systematically characterize mixture interactions at OSNs in their native environment using two-photon imaging of calcium responses to odor stimuli. Imaging populations of OSN axon terminals in the olfactory bulb glomerular layer affords excellent signal-to-noise ratio and direct access to information conveyed to the brain. We also adapted a method to chronically image OSN somata in the olfactory epithelium of freely breathing mice, which allows us to bypass any influence of top-down neural feedback and to directly access the result of odor transduction. By delivering odors individually and in binary mixtures at concentrations that varied over three orders of magnitude, we uncover evidence for multistep activation of ORs and for widespread antagonistic interactions. We also find such antagonistic interactions in responses to complex mixtures containing up to 12 components in single OSNs. Our data strongly support a role for mixture suppression of OSN activity as a normalizing mechanism in olfactory stimulus encoding, which could enhance odor information transfer.

## Results

### Antagonism measured in the glomerular layer of the OB

Odor responses in OSNs can be measured with excellent sensitivity in the glomerular layer of the olfactory bulb, where axons of a given receptor type converge, allowing signal averaging. We used OMP-GCaMP3 mice and imaged axonal calcium responses through a cranial window over the dorsal surface of both olfactory bulbs. For two odorants, Methyl tiglate and Isobutyl propionate (Fig. 2a; Table 1), we measured odor responses in glomeruli across a range of concentrations spanning three orders of magnitude. We then made a binary mixture of the two odorants and measured OSN responses at the same glomeruli (Fig. 2b, c). When plotting the responses as a function of concentration, the mixture concentration at each point represents the sum of the two individual odors—this applies to all experiments where concentration series are used. From nine mice, we identified 334 glomeruli that responded to either of the odors, or the mixture of the two. For this odor pair, many glomeruli reached different saturation levels for the same odor (Fig. 2c).

To estimate antagonism at each glomerulus, we first calculated the average response to the three highest odor concentrations of each individual odor (Fig. 2c, d). We then tested whether the observed mixture responses for the same three concentrations were significantly below the linear sum of the response to the mixture components (rank-sum test of trial replicates). Significant mixture suppression was found in 80.5% (277/344) of glomeruli that responded to at least one odor (Fig. 2e, f).

There are, however, two important considerations in interpreting our result. First, it remains possible that the largest calcium signals observed for individual odors may represent the average maximum physiologically-bounded firing rate across all OSNs of a common type. Similarly, the largest signals we observed for individual odors may be bounded by calcium indicator saturation. Given these constraints, it is plausible that the observed mixture responses can never reach the linear sum of the two mixture components and are simply bounded by the largest response observed for either component alone. To account for this possibility, we compared the observed mixture responses and maximum response for either odor. When using this more conservative metric, we found evidence for antagonism in 41.7% (144/344) of the glomeruli (Fig. 2e-f). These data indicate that, even by a conservative measure, antagonism is widespread in OSNs. Lastly, to ensure that our observations were not only relevant to the highest odor concentrations, we measured the fraction of antagonistic interactions on a sliding scale (Fig. 2g). We found that regardless of odor concentration, antagonism was present in a constant fraction of glomeruli. These data cannot be easily explained by one-step competitive binding models, and provide further evidence that odorant- binding to and activation of odor receptors may be decoupled, as predicted by our earlier theoretical work11.

In these experiments, glomerular responses were highly non-overlapping for the odor pair tested. To ensure that antagonism did not arise from unique interactions between these two odors, and to demonstrate generalization of the phenomenon with other odor pairs, we repeated our experiments using another odor pair, Ethyl valerate and Allyl butyrate (Fig. 3a; Table 2). This pair was specifically selected because of their highly overlapping OSN activation (Fig. 2b). Mixture interactions for this pair of odors were in close agreement with those estimated above for the other odor pair. Using the same approach that we previously described, from 226 glomeruli in five mice, we found antagonism in 76.6% (173/226) of glomeruli when comparing the summed responses and 25.7% (58/226) when comparing against the maximum (Fig. 3e, f). Again, the fraction of glomeruli that exhibited mixture suppression was concentration invariant (Fig. 3g).

Across all glomeruli from both odor pairs where saturating responses were observed for an odor, we could then fit response curves using Eq. (1) (see Methods). We used stringent criteria that ensured only well-activated glomeruli were included by imposing thresholds on the mean-squared error from the best fit parameters. When comparing the fits from glomerular responses to the odors used in Fig. 2 (Fig. 4a; also see Table 1), the distribution and mean Hill coefficients were similar (2.67 ± 0.16, n = 48 for Methyl tiglate and 2.53 ± 0.32, n = 22 for Isobutyl propionate; P > 0.05, Rank-sum test; Fig. 4b). Early experiments with synthetic calcium indicators or intrinsic imaging of glomeruli estimated Hill coefficients to be close to 1, but more recent experiments appear to measure higher values9,10. We then compared the Hill coefficients obtained from the curves fit to the mixture responses (mean = 2.21 ± 0.16, n = 30; Fig. 4d) with those from each of the components alone. We found no differences in any of the comparisons (P > 0.05, Rank-sum test).

When we fit the glomerular responses to the odor pair used in Fig. 3 (Fig. 4c; also see Table 2), the distribution and the means of the Hill coefficients were remarkably similar (2.36 ± 0.19, n = 44 for Ethyl valerate and 2.29 ± 0.21, n = 39 for Allyl butyrate; P > 0.05, Rank-sum test; Fig. 4d), and comparable to those for the odor pair discussed above. We again compared the Hill coefficients extracted from the data fit to the glomerular mixture responses (mean = 2.26 ± 0.22, n = 32; Fig. 4d) to each of the Hill coefficient distributions from the mixture components and found no differences across all comparisons (P > 0.05, Rank-sum test).

### Imaging odor responses in individual OSNs

Our glomerular imaging results suggest that mixture suppression through antagonism is widespread. However, an alternative mechanism that could produce such an effect may operate through GABAb- or D2-mediated suppression of OSN axon terminals30,31,32,33.

To avoid circuit interactions between different OSNs, we directly imaged OSN somata in situ in the olfactory epithelium34, where there is no evidence for efferent modulation that is directly linked to olfactory circuitry35. We first characterized the odor tuning and response dynamics of individual cells to determine whether the response properties of sensory neurons in the dorsal recess (zone 1) of the olfactory epithelium36,37 are congruent with those observed in glomeruli in the dorsal olfactory bulb, and to ensure that we sample from a heterogeneous population of OSNs. We used a panel of 32 monomolecular odors (see Table 3) that activate glomeruli on the dorsal olfactory bulb across a wide range of densities (Fig. 5a, d, f). We then imaged individual cells in the epithelium (Fig. 5b) and compared their tuning to glomeruli in the olfactory bulb. Across odors, there was a strong relationship between the fraction of activated glomeruli and individual OSNs (r = 0.750, P < 0.0001, Fig. 5e). We found a similar relationship when we compared the mean response magnitude at both imaging sites across all glomeruli and somata (r = 0.598, P < 0.003, Fig. 5g). These data provide new functional evidence that support anatomical data for a conserved zonal organization between the olfactory epithelium and the olfactory bulb37.

We also found that the calcium response kinetics across all OSNs in a field of view are highly diverse, and that the response waveform for individual cells is remarkably stable across trials of the same odor when respiration is stabilized (Supplementary Fig. 1). Together these data suggest that a diverse and heterogeneous array of receptor subtypes are present within a relatively restricted patch of olfactory epithelium.

### Antagonism in individual sensory neurons

Given that OSNs in the dorsal recess share many of the same odor tuning characteristics as dorsal olfactory bulb glomeruli, we tested whether antagonism could be observed at the single-cell level using the odor pair we used for imaging glomerular responses. We repeated the experiment described in Fig. 2 and collected data from 964 individual sensory neurons using the odor pair, Methyl tiglate and Isobutyl propionate (Table 2). We again detected nonlinear interactions between the odors (Fig. 6b, c). Mixture-suppression was readily observed in individual cells (54.6% (527/964) for sum comparison and 22.0% (212/964) maximum comparison; Fig. 6e, f). The fraction of cells that showed mixture suppression was smaller than in the glomerular data, which may be attributed to noise from two sources. First, the measured signal at glomeruli represents the population response across all ~10,000 OSNs of a common subtype, compared with single OSN cell bodies in the epithelium imaging. The second source of noise may arise from image acquisition, through the number of pixels contributing to the measured signal. While we typically imaged 10–20 densely packed glomeruli per field of view, in the epithelium, we imaged dozens to hundreds of sparsely arranged cells, thereby substantially reducing the number of pixels contributing to our signal.

To ensure that our observations did not arise from more noisy measurements at the single-cell level, we measured the fraction of mixture responses that exceeded the linear prediction of the mixture components, a phenomenon known as synergy. We first measured the fraction of mixture responses that exceeded maximum of either of the mixture components. We found only 11.8% (114/964) of OSNs exceeded the maximum response of either component (Fig. 6e, f); however, it should be noted that for synergy to occur the measured OSN response should exceed the linear summation of the response of both odor components. In our dataset, we found only 5/964 cells where the mixture response was significantly greater than the sum of the mixture components. Both fractions are well below those we measure for antagonism and consistent with previous reports that antagonism is the predominant mixture interaction when compared with synergy38.

We also fit the odor responses in individual OSNs and compared the extracted Hill coefficients to those obtained by imaging populations of OSN at their axon terminals (Supplementary Fig. 2). We predicted that the fits for individual OSNs may be sharper, and therefore have larger Hill coefficients than glomerular fits because measurements from glomeruli average hundreds of OSNs that may have heterogenous odor sensitivities. However, when we made these comparisons, we found no difference between individual OSN and glomerular Hill coefficients for the same odors (Methyl tiglate, OSN mean = 3.31 ± 0.76, n = 6, glomerular mean = 2.67 ± 0.16, n = 48; Isobutyl propionate, OSN mean = 2.65 ± 0.16, n = 42, glomerular mean = 2.53 ± 0.32, n = 22; MT + IP mixture, OSN mean = 2.67 ± 0.37, n = 48, glomerular mean = 2.26 ± 0.22, n = 32; P > 0.05 for all comparisons, Rank-sum test).

### Antagonism in complex odor blends

After establishing that antagonism is a prevalent feature of odor mixture encoding in OSNs, we next wanted to understand the relationship between the complexity of an odor blend (that is, the number of elements in the mixture) and non-linearities in OSN responses. We imaged single OSN responses to 16 monomolecular odors (odor index 1–16 in Table 3), as well as random blends of 2, 4, 8, or 12 odors from this panel (Supplementary Table 1; Fig. 7c). When constructing complex odor mixtures, the concentration of each odor component was the same as when delivered alone. From the OSN responses to each of the single odors, we then made linear predictions for each of the mixture complexities. Our choice of odors for this experiment was motivated by several previous studies that demonstrate their efficacy and diversity in activating OSNs that presumably signal through GPCR signaling pathways39,40,41, as well as past behavioral work14,42, which demonstrates that mixtures of these odors can be perceived and distinguished by mice despite their overlapping OSN activity patterns.

For all mixture complexities, we routinely observed OSN responses that were far less than the linear prediction made by the summation of the individual mixture components (Fig. 7d). However, OSNs may never achieve the response predicted by linear summation, due to either firing rate or indicator saturation. To account for this possibility, for each OSN, we fit the data across all mixture complexities with a sigmoid with an initial slope of 1 and reaching an asymptote at the top 0.05 quantile of all observed mixture responses (n = 100). For each cell, we then calculated the deviation of each mixture response from the sigmoidal fit estimating the maximum response, or the linear prediction. We considered mixture responses that fell below the sigmoidal fit to be representative of antagonism (black line; Fig. 7d), while response that fell above the sigmoid represented synergistic interactions.

We observed nonlinear mixture responses in OSNs at all mixture complexities, although both the frequency and magnitude of such interactions increased with mixture complexity (Fig. 7f, n = 1800, n = 1596, n = 1861, n = 1948 OSN-odor mixture pairs, for 2-, 4-, 8-, and 12-part mixtures, respectively). For this analysis, we only considered OSN-mixture pairs where the predicted OSN response was >0 ΔF/F. Importantly, mixture suppression was not the result of large trial-to-trial variability where a single trial strongly influenced the mean as the error for each mixture complexity was well below the sigmoidal fit to the data (Supplementary Fig. 3). For this analysis, we pooled the OSN responses for each mixture complexity and subsampled the distributions to test the significance of the observed nonlinearities against a null distribution. 200 random responses for each mixture complexity were selected and compared with distributions with a vanishing mean and the same standard deviation using a Kolmogorov–Smirnov test. This process was then repeated 100,000 times. Distributions of the P values obtained from these comparisons at each mixture complexity are shown on the right in Fig. 7f. For each mixture complexity we found >99% of all bootstrapped comparisons resulted in P values < 0.05.

To even more rigorously test our data, we performed an additional analysis where we compared OSN responses to odor mixtures against their maximum response to any of the mixture components alone (Supplementary Fig. 4). In close agreement with our prior analysis, we again found that likelihood, and the magnitude of mixture suppression in OSNs was related to the number of odor components contained within a mixture.

To exclude the possibility that a small number of cells most strongly contribute to the antagonism we observe, we normalized all measurements from each cell to the asymptote of the sigmoidal fit to the data and repeated our analysis. Here, we again observe that antagonism increases with mixture complexity (Fig. 7g). As a further confirmation of our finding, we repeated this experiment in OSN axon terminals in the glomerular layer. We collected data from 39 glomeruli in four mice. We again observed strongly nonlinear mixture interactions that increased with mixture complexity (Supplementary Fig. 5; n = 673, n = 637, n = 733, n = 750 glomeruli-odor mixture pairs, for 2, 4, 8, and 12-part mixtures respectively). Together, our results demonstrate that input to the olfactory system may be normalized through mixture suppression of OSN activity.

## Discussion

It has been known for some time that individual odorant receptors can respond to multiple odors23,43, so a reasonable question is how these receptors will respond to natural stimuli, which are almost always mixtures of odors. Odor mixture interactions have been described behaviorally for many decades44,45,46. One form of interaction is where binary (and larger) mixtures tend to have reduced perceived intensity (suppression). Odors can also be masked or harder to identify in mixtures, which could be due to reduced sensation of specific components. The origin of this perceptual odor suppression in mixtures is uncertain. Some of it could arise at the periphery, including at the receptors themselves. We recently proposed a simple model based on the two-step activation of odorant receptors, and noted that even a modest decorrelation of binding affinity and activation efficacy across odors could lead to antagonistic interactions, which will lead to mixture suppression11. In other words, odors that bind the most tightly to odor receptors may not strongly activate the odor receptor complex and the downstream signaling cascade that ultimately generates OSN output.

Here, we have used direct imaging of OSN somata, as well as more conventional glomerular imaging, to show that mixture suppression (by inference antagonistic interactions) is widespread. We adapted a method first reported by Iwata et al.34, to directly image the somata of OSNs in vivo, in freely breathing mice. We obtained responses of OSNs to a diverse array of odorants and found that the statistics of population responses were similar to that obtained for glomeruli. Interestingly, inhibitory responses were more easily detected in glomeruli than in OSN cell bodies, which might reflect glomerular averaging of heterogeneous responses of OSNs47. This OSN imaging method is likely to be valuable for understanding the fundamental properties of sensory coding and modulation at the periphery, especially with chronic imaging.

Previous studies using binary mixture stimuli have offered conflicting evidence for nonlinear interactions17,24,48,49. Evidence for relatively linear summation of odor responses to mixtures has been offered by studies that have imaged glomerular responses at low spatial and temporal resolution48,49,50, but these studies did not systematically vary the concentration of odorants. Extracellular spike recordings from single rat OSNs have indicated that a simple competitive binding model can explain the responses in about half the cases17. In the rest of the cases, more complex interactions including antagonism (competitive or non-competitive) and masking were needed to explain the observations17. Insect olfactory receptors also exhibit such nonlinear interactions51,52. Recently, we offered a principled explanation for all these different effects using a model that decouples the binding affinity of odorants to receptors and their activation efficiency11. Then, depending on the statistics of these properties in the repertoire of odorants encountered by an animal, the population of odorant receptors could exhibit varying degrees of antagonism or synergy.

In this work, we used binary mixtures and concentration variation over three orders of magnitude, to find that suppressive mixture interactions were remarkably widespread. We first demonstrated mixture suppression with glomerular imaging. Since glomerular responses are averages of many hundreds of OSN axons, they will overlook any heterogeneity. Glomerular responses may also be influenced by lateral interactions within the olfactory bulb networks, particularly due to GABAb-mediated presynaptic inhibition that can alter OSN calcium signals33,53. However, there is strong evidence that GABAb-mediated reduction of OSN responses are largely intraglomerular30, which will simply serve as an automatic gain control, hence insensitive to the identity of the odorant. As an additional check, we chose two distinct odor mixtures with differing extents of overlap in glomerular activation. We found remarkably similar mixture effects for both pairs of odorants, which indicates that lateral interactions among glomeruli are unlikely to affect our conclusions.

To circumvent circuit and feedback interactions, we imaged OSNs directly and found that binary mixture suppression was just as widespread as observed with glomerular imaging. Since these interactions were observed at the earliest stages of odor encoding, in OSNs, the most likely source of mixture interactions is odorant receptors themselves. Evidence of antagonistic interactions in OSNs has been reported previously16,17,18,54,55, and we advance these prior observations by providing evidence for widespread occurrence of this phenomenon. Going beyond binary mixtures, we used odor blends of increasing complexity, up to 12 components, to ask how mixture responses compared with the expected linear summation of individual responses (below saturation). This allowed us to ask how OSNs respond to more complex mixtures. It has been previously hypothesized that such nonlinearities in peripheral coding may be a mechanism to increase precision and specificity of encoding complex odor stimuli11,12,17,54. Indeed, using more naturalistic odor delivery conditions, we found that the degree of mixture suppression was greater with increasing number of components, which is expected from statistical consideration of antagonistic interactions11. This feature can be rationalized as increasing normalization of population responses, which we have shown previously to allow greater information transfer about odor identity, when saturation threatens degradation of information11.

An important point regarding the odors we used in this study is that many of the odors share common molecular features (e.g., ester groups). A reasonable assumption is that there should be some molecular overlap between odors for antagonism to occur, due to the necessity of interactions between odor molecules and the binding pocket of olfactory receptors. However, in our panel we did include several odors that did not contain an ester group. When these non-ester odors were delivered as binary mixtures, in combination with ester containing odors, mixture suppression was still observed (Supplementary Fig. 6). These additional data, along with our previous observation that narrowly tuned ORNs show strong mixture suppression (Fig. 7d), argue that antagonism is indeed a fundamental feature of mixture encoding in ORNs and is not a unique feature restricted to those that are sensitive to a single functional group.

Beyond our theoretical observations, there are other more recent experimental studies that report related effects18,19,20. Our work is unique in the following regard: (1) we report responses from a large set of OSNs in intact, freely breathing mice, responding to vapor phase odorants inhaled physiologically, (2) we report fast, real-time responses with a variety of odorants and using both OSN and glomerular imaging and (3) we report responses to mixtures of high complexity (up to 12 odors), well beyond just binary mixtures.

## Methods

### Experimental model and subject details

Adult heterozygous OMP-GCaMP3 mice60 of both sexes were used in this study. All animals were produced from a breeding stock maintained within Harvard University’s Biological Research Infrastructure. All animals were between 20 and 30 g before surgery and singly housed following any surgical procedure. Animals were 2–6 months old at the time of the experiments. All mice used in this study were housed in an inverted 12 h light cycle and fed ad libitum. Animals were housed at 22 ± 1 °C at 30–70% humidity. All the experiments were performed in accordance with the guidelines set by the National Institutes of Health and approved by the Institutional Animal Care and Use Committee at Harvard University.

### Olfactory epithelium thinned skull procedure

Mice were anesthetized using the same procedure and all pre-surgical methods through head plate implantation are the same as the craniotomy. The cranial bones over the olfactory epithelium were thinned with a dental drill and blade until transparent34. The thinned area of skull was then covered with cyanoacrylate adhesive (Loctite) and a class coverslip was implanted in the adhesive. Dental cement was then used to form a well over the thinned section of skull. Following a recovery period, the thinned bone procedure allowed for chronic imaging of the epithelium for several weeks.

### Multiphoton Imaging

A custom-built two-photon microscope was used for in vivo imaging. Fluorophores were excited and imaged with a water immersion objective (20×, 0.95 NA, Olympus) at 920 nm using a Ti:Sapphire laser (Mai Tai HP, Spectra-Physics). Images were acquired at 16-bit resolution and 4–8 frames/s. The pixel size was 0.6 μm OSN somata imaging and 1.2–2.4 μm for imaging glomeruli. Fields of view ranged from 180 × 180 μm in the epithelium to 720 × 720 μm in the glomerular layer. The point-spread function of the microscope was measured to be 0.51 × 0.48 × 2.12 μm. Image acquisition and scanning were controlled by custom-written software in LabView (National Instruments). Emitted light was routed through two dichroic mirrors (680dcxr, Chroma and FF555- Di02, Semrock) and collected by a photomultiplier tube (R3896, Hamamatsu) using filters in the 500–550 nm range (FF01–525/50, Semrock).

### Odor stimulation

Monomolecular odorants (Sigma or Penta Manufacturing) were used as stimuli and delivered by custom-built 16 channel olfactometers controlled by custom-written software in LabView61. For binary mixture experiments, the initial concentration series for each odor was between 0.08 and 80% (v/v) in mineral oil and further diluted 16 times with air. For all experiments, the airflow to the animal was held constant at 100 mL/min and odors were injected into a carrier stream. The relative odor concentration was measured by a photoionization detector (PID; Aurora Scientific) and normalized to the largest detected signal for each odor. To create mixtures, air-phase dilution was used, and the total concentration of each odor was held constant. The measured mixture signal in the PID was nearly a perfect linear summation of the signal measured for each odor alone (Supplementary Fig. 7). For all experiments, odors were delivered 2–6 times each.

For experiments characterizing the odor tuning of olfactory epithelium, the odor panel consisted of: (1) Ethyl tiglate (2) Allyl tiglate (3) Hexyl tiglate (4) Methyl tiglate (5) Isopropyl tiglate (6) Citronellyl tiglate (7) Benzyl tiglate (8) Phenylethyl tiglate (9) Ethyl propionate (10) 2-Ethyl hexanal (11) Propyl acetate (12) 4-Allyl anisole (13) Ethyl valerate (14) Citronellal (15) Isobutyl propionate (16) Allyl butyrate (17) Methyl propionate (18) Pentyl acetate (19) Valeric acid (20) (+)Carvone (21) (−)Carvone (22) 2-Methoxypyrazine (23) Isoeugenol (24) Butyl acetate (25) Valeraldehyde (26) Isoamyl acetate (27) Methyl valerate (28) Octanal (29) 2-Hexanone (30) Methyl butyrate (31) 2-Heptanone (32) Acetophenone. See Supplementary Fig. 7 for PID measurements. For experiments measuring complex mixture responses in the olfactory epithelium, odors 1–16 were used from the panel above. Additional odor information is available in Table 3 and Supplementary Table 1.

### Data analysis

Images were processed using both custom and available MATLAB (Mathworks) scripts. Motion artifact compensation and denoising was done using NoRMcorre64. For experiments imaging OSN axon terminals in the olfactory bulb, regions of interest (ROIs) were manually selected by outlining glomeruli in maximum projection images. For epithelium imaging, the CaIMaN CNMF pipeline65 was used to select and demix ROIs. ROIs were further filtered by size and shape to remove merged cells. For all mixture experiments, the peak ΔF/F signal was calculated by finding the peak signal following odor onset and averaging with the two adjacent points. The mean ΔF/F signal in the 20 frames following odor onset was used for odor tuning experiments. To account for changes in respiration frequency and anesthesia depth, correlated variability across replicates was corrected for14. Thresholds for classifying responding ROIs were determined from a noise distribution of blank (no odor) trials from which three standard deviations were used for responses. Across all datasets, only ROIs with at least one significant odor response were included for further analysis. Measurements of binary mixture non-linearities used individual trial replicates of the three highest odor concentrations used in each experiment.

### Data fitting

We further analyzed the response curves from 344 glomeruli corresponding to Methyl tiglate, Isopropyl propionate, and their mixture, 226 glomeruli corresponding to Ethyl valerate, Allyl butyrate, and their mixture, and 964 ORNs corresponding to Methyl tiglate, Isopropyl propionate and their mixture. The observed Ca2+ fluorescence response R(c) was fit using a sigmoid function against log odor concentrations delivered over three orders of magnitude using the equation:

$$R\left( c \right) \,=\, b + R_{\mathrm{max}}/(1 + e^{ - n({\rm{log}}\,c - {\rm{log}}{\rm{K}})}).$$
(1)

The four free parameters that are fit to the data are b, Rmax, Κ and n. Here b is interpreted as the baseline response for blank odors, Rmax is the response above baseline at saturating concentrations, Κ (commonly referred to as the dissociation constant) is the reciprocal of the binding affinity and n is the Hill coefficient. To account for noise and variability in the data, we used four criteria to select the fitted curves (all units are ∆F/F): (1) The total squared deviation of the data from the best fit sigmoid shouldn’t be too large. We used a threshold of 0.1 to ensure rejection of highly noisy data. (2) The maximum value over all concentrations should be >0.4 ∆F/F, (3) Binding affinity should be within the range of the concentrations used and (4) the hill coefficient cannot be below 0.75 or above 6 to exclude unreasonably sharp fits.

### Reporting summary

Further information on research design is available in the Nature Research Reporting Summary linked to this article.

## Data availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.

## Code availability

The code used for analysis and figure generation of this study are available from the corresponding author upon reasonable request.

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## Acknowledgements

This work was partly supported by grants from the NIH (R01 DC014453 & DC016289) to VNM. JDZ was supported by NIH Fellowships F32 DC015938 & K99 DC017754. GR was partially supported by the NSF-Simons Center for Mathematical & Statistical Analysis of Biology at Harvard (award number #1764269) and the Harvard Quantitative Biology Initiative. We also acknowledge support from the following grants, which facilitated critical discussions at KITP, Santa Barbara: NSF (PHY-1748958), NIH (R25 GM067110), and the Gordon and Betty Moore Foundation (2919.01). We would also like to thank members of the Murthy Lab for helpful discussions.

## Author information

Authors

### Contributions

J.D.Z., G.R., M.V., and V.N.M. designed the experiments. J.D.Z. acquired the data. J.D.Z. and G.R. analyzed the data. J.D.Z. and V.N.M. wrote the paper with inputs from all authors.

### Corresponding authors

Correspondence to Joseph D. Zak or Venkatesh N. Murthy.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

Peer review information Nature Communications thanks Patricia Duchamp-Viret, Matthew Smear and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. Peer reviewer reports are available.

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

## Rights and permissions

Reprints and Permissions

Zak, J.D., Reddy, G., Vergassola, M. et al. Antagonistic odor interactions in olfactory sensory neurons are widespread in freely breathing mice. Nat Commun 11, 3350 (2020). https://doi.org/10.1038/s41467-020-17124-5

• Accepted:

• Published:

• ### Brief olfactory learning drives perceptive sensitivity in newborn rabbits: New insights in peripheral processing of odor mixtures and induction

• Patricia Duchamp-Viret
• , Jiasmine Boyer
• , Florian La Villa
•  & Gérard Coureaud

Physiology & Behavior (2021)

• ### Odor coding in the mammalian olfactory epithelium

• Smija M. Kurian
• , Rafaella G. Naressi
• , Diogo Manoel
• , Ann-Sophie Barwich
• , Bettina Malnic
•  & Luis R. Saraiva

Cell and Tissue Research (2021)

• ### Mixture interactions at mammalian olfactory receptors are dependent on the cellular environment

• Elizabeth A. Corey
• , Sergei Zolotukhin
• , Barry W. Ache
•  & Kirill Ukhanov

Scientific Reports (2021)

• ### Mixture Coding and Segmentation in the Anterior Piriform Cortex

• Sapir Penker
• , Tamar Licht
• , Katharina T. Hofer
•  & Dan Rokni

Frontiers in Systems Neuroscience (2020)

• ### A Systematic Framework for Olfactory Bulb Signal Transformations

• Thomas A. Cleland
•  & Ayon Borthakur

Frontiers in Computational Neuroscience (2020)


text:
# Math Help - Demand and Total Cost Curves

1. ## Demand and Total Cost Curves

I am given that:
Q=100-P
TC = 1/3Q³-7Q²+111Q+50

Find output and price that maximise profit and then calculate profit. Ensure you have found a maximum and checked for boundary maxima.

So far I have done this:

Rearranged Q so that
P= -Q+100

Q*P = TR = -Q+100 (Q) = -Q²+100Q

Profit = TR-TC= -Q²+100Q-1/3Q³-7Q²+111Q+50
=1/3Q³-6Q²-11Q+50

First Derivative = -Q²-12Q-11
(-Q-11)(Q+1)
Therefore Q=11 or Q=-1

Then Used Second Deriv to work out Min/Max:
=-2Q-12
When Q=11, Maximum
When Q=-1, Maximum

Now I am unsure If I have made a mistake up to this point however I sort of come stuck here. Does anyone know if I have made any mistakes and what the next step would be?

My thought is that you cant have a negative quantity and therefore Q must = 11. When Q = 11, P=89

and then i was just going to plug the figures into the Profit formula and see what i get?!

2. Assuming your algebra is correct, You are right to discount Q=-11.

a maximum can occur at either:

Q=0
Q=100 (corresponds to P=0)
Or somewhere in between.

Your differenciation has given you the maximum profit in the "somewhere in between" range. "Check for boundary maxima" means check that this gives higher profits than Q=0 or Q=100.

format: plain


text:
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# How many positive integers less than 50 have a reminder 01 w

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Intern
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How many positive integers less than 50 have a reminder 01 w  [#permalink]

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31 Jul 2014, 10:16
3
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Difficulty:

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Question Stats:

58% (01:17) correct 42% (01:22) wrong based on 325 sessions

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How many positive integers less than 50 have a reminder 01 when divided by 3?

A)13
B)14
C)15
D)16
E)17

I did:
Since to have a reminder when when divided by 3, the must be one unit greater than the multiples of 3 I divided 50/3 and get 16 numbers multiples of 3
( from 3 to 48) so I got from 4 to 49 number with the reminder = 1.

What am I missing?
Thank you!
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Joined: 02 Sep 2009
Posts: 59020
Re: How many positive integers less than 50 have a reminder 01 w  [#permalink]

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31 Jul 2014, 10:37
4
2
Reni wrote:
How many positive integers less than 50 have a reminder 01 when divided by 3?

A)13
B)14
C)15
D)16
E)17

I did:
Since to have a reminder when when divided by 3, the must be one unit greater than the multiples of 3 I divided 50/3 and get 16 numbers multiples of 3
( from 3 to 48) so I got from 4 to 49 number with the reminder = 1.

What am I missing?
Thank you!

1 also gives the remainder of 1 when divided by 3. So, there are total of 17 numbers.

Please tag the questions properly. This is a REMAINDERS question!

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Re: How many positive integers less than 50 have a reminder 01 w  [#permalink]

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31 Jul 2014, 14:24
1
Reni wrote:
How many positive integers less than 50 have a reminder 01 when divided by 3?

A)13
B)14
C)15
D)16
E)17

I did:
Since to have a reminder when when divided by 3, the must be one unit greater than the multiples of 3 I divided 50/3 and get 16 numbers multiples of 3
( from 3 to 48) so I got from 4 to 49 number with the reminder = 1.

What am I missing?
Thank you!

IMO you have to put the question this way to avoid any confusions in remainder questions
x= 3y + 1 ( x all the numbers that satisfies the rule)
since x is positive, you will have to start with y= 0 (which will give us x=1) till y= 16 (which will give us x=49)
this will give you = 16 - 0 + 1 =17 positive integers satisfying the rule
In your reasoning u started with 4 rather than 1
hope that helps
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Re: How many positive integers less than 50 have a reminder 01 w  [#permalink]

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01 Aug 2014, 00:20
1
Closest number to 50 which is divisible by 3 = 51

$$\frac{51}{3} = 17$$

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How many positive integers less than 50 have a reminder 01 w  [#permalink]

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01 Aug 2014, 19:59
2
Last number in series: 49
First number in series: 1

Spacing b/w each number: 3

Hence counting the number of such numbers in series: $$((49-1)/3)+1$$$$=17$$
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Re: How many positive integers less than 50 have a reminder 01 w  [#permalink]

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25 Sep 2014, 17:54
between 1 - 49, we have 1, 4, 7...49.

total = (49-1)/3 + 1 = 16+1 = 17

Reni wrote:
How many positive integers less than 50 have a reminder 01 when divided by 3?

A)13
B)14
C)15
D)16
E)17

I did:
Since to have a reminder when when divided by 3, the must be one unit greater than the multiples of 3 I divided 50/3 and get 16 numbers multiples of 3
( from 3 to 48) so I got from 4 to 49 number with the reminder = 1.

What am I missing?
Thank you!

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Joined: 14 Sep 2014
Posts: 86
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Re: How many positive integers less than 50 have a reminder 01 w  [#permalink]

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25 Sep 2014, 20:21
Reni wrote:
How many positive integers less than 50 have a reminder 01 when divided by 3?

A)13
B)14
C)15
D)16
E)17

I did:
Since to have a reminder when when divided by 3, the must be one unit greater than the multiples of 3 I divided 50/3 and get 16 numbers multiples of 3
( from 3 to 48) so I got from 4 to 49 number with the reminder = 1.

What am I missing?
Thank you!

you just missed to take number 1, which will give remainder 1.
Also its remainder not reminder
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How many positive integers less than 50 have a reminder 01 w  [#permalink]

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26 Sep 2014, 00:21
Closest number to 50 which is divisible by 3 = 51

\frac{51}{3} = 17

This is real time saving trick but how reliable is it? can we adopt it for all such category question?
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Re: How many positive integers less than 50 have a reminder 01 w  [#permalink]

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26 Sep 2014, 00:34
taleesh wrote:
Closest number to 50 which is divisible by 3 = 51

\frac{51}{3} = 17

This is real time saving trick but how reliable is it? can we adopt it for all such category question?
_________________

For same type of questions, yes it will work for sure

Only we have to filter our "same type questions" per se
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How many positive integers less than 50 have a reminder 01 w  [#permalink]

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Updated on: 29 Jan 2015, 23:28
1
1
Can also be solved by arithmetic series formula. fool-proof against leaving out edge cases.

a1 = 1, an = 49, d = 3 (1,4,7,10..)

an = a1 + (n-1)d
49 = 1 + (n-1) * 3

n = (48/3) + 1 = 17

Originally posted by Gbv11 on 09 Dec 2014, 13:17.
Last edited by Gbv11 on 29 Jan 2015, 23:28, edited 1 time in total.
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Re: How many positive integers less than 50 have a reminder 01 w  [#permalink]

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10 Dec 2014, 04:43
1 divided by 3 also gives remainder 1. so we have a series here starting with 1 and ending with 49. since 49 is the last number below 50 which leaves remainder 1 when divided by 3.
so equation becomes let N be the number required so 49= 1 +(N-1)*3 N=17.
Hope this explanation helps.
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Re: How many positive integers less than 50 have a reminder 01 w  [#permalink]

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02 Jan 2015, 01:05
Notice here first we should find a pattern in which whenever we divide the number less than 50 we obtain reminder 1.

the first integer in this pattern is 1 itself . because when we divide 1 by 3 it gives reminder 1. the next integer is 1+3 = 4 and the next will be 4+3 =7 and so on

on the other hand the last integer in this pattern will be 49 as the problem says that integer less than 50 and when we divide 49 by 3 gives us reminder 1.

so we have the pattern as follows: 1, 4, 7,.......,49 Now in counting the number of member of one progression we should subduct the last item from the first and divide it by the common difference between its members plus 1 .

last number :49
First number :1

common difference: 3

so we have: 49-1[fraction]3= 16 + 1 = 17 so Option E.....
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Re: How many positive integers less than 50 have a reminder 01 w  [#permalink]

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13 May 2015, 04:59
Bunuel wrote:
Reni wrote:
How many positive integers less than 50 have a reminder 01 when divided by 3?

A)13
B)14
C)15
D)16
E)17

I did:
Since to have a reminder when when divided by 3, the must be one unit greater than the multiples of 3 I divided 50/3 and get 16 numbers multiples of 3
( from 3 to 48) so I got from 4 to 49 number with the reminder = 1.

What am I missing?
Thank you!

1 also gives the remainder of 1 when divided by 3. So, there are total of 17 numbers.

Please tag the questions properly. This is a REMAINDERS question!

from 1 to 49 , there are( 49-1)/3 = 16 numbers. but we need to add 1 more because we count the two end.
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Re: How many positive integers less than 50 have a reminder 01 w  [#permalink]

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13 Jun 2015, 13:38
I also tried to find a pattern.

So, starting with 1, it leaves a remainder of 1.
2 leaves a remainder of 2.
4 leaves a remainder of 1.
5 leaves a remainder of 2.
7 leaves a remainder of 1.

If it hadn't occured to us from the begining that the numbers should increase by 3, it should occur by now.

Then you can just count them, since they are too few, or use the countring method outlined above (last minus first, divided by the increment, plus one because it is inclusive): (49-1)/3 + 1 = 17.

If you counted you would end up with sth like this:
1-4-7-10-13-16-19
22-25-28-31-34-37-40
43-46-49
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Re: How many positive integers less than 50 have a reminder 01 w  [#permalink]

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20 Jul 2017, 16:31
1
Reni wrote:
How many positive integers less than 50 have a reminder 01 when divided by 3?

A)13
B)14
C)15
D)16
E)17

We have a set of evenly spaced integers. The first integer less than 50 that has a remainder of 1 when divided by 3 is 1, and the largest number meeting the same criterion is 49.

Thus, there are (49 - 1)/3 + 1 = 17 numbers.

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Re: How many positive integers less than 50 have a reminder 01 w  [#permalink]

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02 Aug 2018, 18:09
Reni wrote:
How many positive integers less than 50 have a reminder 01 when divided by 3?

A)13
B)14
C)15
D)16
E)17

let x=number of integers
1+3(x-1)=49
x=17
E
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# HP90

A general $\text{2D-} hp\text{-adaptive}$ Finite Element (FE) implementation in Fortran 90 is described. The implementation is based on an abstract data structure, which allows to incorporate the full $hp$-adaptivity of triangular and quadrilateral finite elements. The $h$-refinement strategies are based on $h2$-refinement of quadrilaterals and $h4$-refinement of triangles. For $p$-refinement we allow the approximation order to vary within any element. The mesh refinement algorithms are restricted to 1-irregular meshes. Anisotropic and geometric refinement of quadrilateral meshes is made possible by additionally allowing double constrained nodes in rectangles. The capabilities of this $hp$-adaptive FE package are demonstrated on various test problems.

## Keywords for this software

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ISSN:2164-6376 (print)
ISSN:2164-6414 (online)
Discontinuity, Nonlinearity, and Complexity

Dimitry Volchenkov (editor), Dumitru Baleanu (editor)

Dimitry Volchenkov(editor)

Mathematics & Statistics, Texas Tech University, 1108 Memorial Circle, Lubbock, TX 79409, USA

Email: dr.volchenkov@gmail.com

Dumitru Baleanu (editor)

Cankaya University, Ankara, Turkey; Institute of Space Sciences, Magurele-Bucharest, Romania

On the Extreme Points of the Unit Ball in the Space of Solenoidal Vector Measures on the Plane

Discontinuity, Nonlinearity, and Complexity 9(4) (2020) 525--528 | DOI:10.5890/DNC.2020.12.005

Nikolay A. Gusev

Moscow Institute of Physics and Technology, 9 Institutskiy per., Dolgoprudny, Moscow Region, 141700, Russia

Abstract

We consider the space of finite divergence-free Borel vector measures on~$\mathbb R^d$, endowed with the total variation norm. For $d=2$ we present a characterization of the extreme points of the unit ball in this space. This allows one to decompose (for $d=2$) any finite divergence-free vector measure into measures induced by closed Lipschitz curves. The results are based on a joint work with P. Bonicatto.

References

1.  [1] Stepanov, E. and Trevisan, D. (2017), Three superposition principles: Currents, continuity equations and curves of measures, { Journal of Functional Analysis}, 272(3), 1044-1103.
2.  [2] Ambrosio, L. and Crippa, G. (2008), { Existence, Uniqueness, Stability and Differentiability Properties of the Flow Associated to Weakly Differentiable Vector Fields}, pages 3-57, Springer Berlin Heidelberg, Berlin, Heidelberg.
3.  [3] Bianchini, S., Bonicatto, P., and Gusev, N.A. (2016), Renormalization for autonomous nearly incompressible {BV} vector fields in two dimensions, { {SIAM} Journal on Mathematical Analysis}, 48(1), 1-33.
4.  [4] Smirnov, S.K. (1993), Decomposition of solenoidal vector charges into elementary solenoids, and the structure of normal one-dimensional flows, { Algebra i Analiz}, 5, 206-238.
5.  [5] Paolini, E. and Stepanov, E. (2012), Decomposition of acyclic normal currents in a metric space, { Journal of Functional Analysis}, 263(11), 3358-3390.
6.  [6] Paolini, and Stepanov, E. (2013), Structure of metric cycles and normal one-dimensional currents, { Journal of Functional Analysis}, 264(6), 1269-1295.
7.  [7] Phelps, R.R. (2001), { Lectures on Choquet's Theorem}, Lecture Notes in Mathematics, Springer Berlin Heidelberg.
8.  [8] Bonicatto, P. and Gusev, N.A. (2019), On the structure of divergence-free measures on the plane, { In preparation}.
9.  [9] Ambrosio, L., Caselles, V., Masnou, S., and Morel, J.M. (2001), Connected components of sets of finite perimeter and applications to image processing, { Journal of the European Mathematical Society}, 3(1), 39-92.
10.  [10] Bianchini, S. and Gusev, N.A. (2016), Steady nearly incompressible vector fields in two-dimension: Chain rule and renormalization, { Archive for Rational Mechanics and Analysis}, 222(2), 451-505.


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# What is the difference between Cross Correlation and Correlation Matrix

As the title implies I am having difficulty differentiating between the correlation matrix and the cross correlation matrix with reference to time series data. Could anyone enlighten me please? Is the only difference that the cross correlation uses a sliding window to calculate the correlation over time whereas the correlation is time independent?

According to wikipedia the cross correlation is a measure of similarity of two series as a function of the displacement of one relative to the other. That is one signal is delayed relative to the other and the correlation between the two is calculated, kind of similar to the autocorrelation but with another signal rather than itself.


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# Harmonic function, existence of a constant

May i ask you for a little help about a problem with harmonic function? It seems to be not that difficult, in a way even intiutively obvious but i don't really know how to show this explicitly.

We have $U\subset \mathbb{R}^{2}$ open, non-empty, connected subset and a harmonic function $u: U\rightarrow \mathbb{R}$ given by $f:=\frac{\partial u}{\partial x} - i \frac{\partial u}{\partial y}$. Let $F$ be a holomorphic function $U\rightarrow \mathbb{C}$ with $F'=f$. Show that there exist an $c\in \mathbb{R}$ such that $\operatorname{Re}(F) = u + c$.

Okay, $f$ is given only through $u$. This means that after the integration of $F$ i will have an expression with the real part of $u(x,y)$.

I would be glad if someone could help me to move on. Thank you in advance!

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I don't even think this is true as stated. I feel as though we may need some connectivity conditions on $U$. If we assume that $U$ is simply connected this is not bad, for example. – Alex Youcis Mar 9 '13 at 23:29
@AlexYoucis, thank you for the correction! $U$ must be open, nonempty and connected. I edited my first post. – Lullaby Mar 9 '13 at 23:32
I have posted an answer! – Alex Youcis Mar 9 '13 at 23:34
Just connected is not enough. You need that $U$ is simply connected, otherwise $F$ might not exist. – mrf Mar 9 '13 at 23:36
@mrf There is no issue if you are given the $f$. But yes, we clearly need simply connected if we want to guarnatee the existence of such an $f$. – Alex Youcis Mar 12 '13 at 16:24

Your formulation is a little strange. I assume that you start with a harmonic function $u$ and then define $f$.

First note that $f$ is holomorphic since it satisfies Cauchy-Riemann's equations. Assuming that $U$ is simply connected (and this assumption is essential), $f$ has a primitive, $F$. Again, using Cauchy-Riemann, it follows that if $F=a+ib$, you have that $a'_x = \operatorname{Re} f = u'_x$ and similarly $a'_y = u'_y$, so using that $U$ is connected, $a = u+c$.

(This is one common way to prove the existence of harmonic conjugates.)

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Hello mrf, I have the same problem as the OP, but I don't understand your answer, specially when you say "using that $U$ is connected, $a = u + c$. How could you use this assumption to conclude $a= u+c$. Sorry if my question sounds stupid, but I'm not used to work with connectedness of sets. – Ale Mar 8 '14 at 10:31
I've also another question, why can you say that f has a primitive because $U$ is connected, does it follow from some definition? – Ale Mar 8 '14 at 10:35
Hi @mrf, if I have a harmonic function with known boundary values, it apparently must be unique. I am thinking that this is true, just from looking at the Poisson integral formula, which gives its values on the interior, from knowing its values on the boundary. However, why is the Maximum Principle needed here to argue the uniqueness of the harmonic function? I've seen it show up on several answers, but I don't see why it's necessary. Thanks professor :-) – User001 Dec 17 '15 at 19:05

Hint: Apply the Cauchy-Riemann equations to conclude that the total derivative of $\text{Re}(F)$ is equal to the total derivative of $u$. Use then connectivity and the Intermediate Value Theorem to conclude that $u$ and $\text{Re}(F)$ differ by a constant.

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# Could the earth's atmosphere have been partially stripped by a passing rogue planet?

I am involved in the fight between palaeontologists and aerodynamicists on the theoretical flight capabilities of the largest pterosaurs (who reached their climax in the Cretaceous and then disappeared at the K/T extinction event).

Is it possible that the earth's atmosphere was reduced at this time by a passing rogue planet or other sufficiently large body?

If so, the atmosphere would have been thicker.

Using the principle of Occam's razor, this one single hypothesis solves the problem of flight in the Mesozoic; the problem of how the dinosaurs could carry such weight; why they all had tails that would be much more use in a thick medium such as water...

The K/T extinction has attracted too many theories about what was 'added' to life on earth such as the comet/asteroid/Deccan Traps theories etc. No-one has looked at what might have been taken away from earth to cause the same effect!

If the earth's atmosphere had been much thicker (far higher pressure/density at the surface) the dinosaurs would have had a much easier time. Losing a huge amount of their atmosphere would kill them off because their 'effective weight' would have been doubled almost overnight.

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Though this is definitely an interesting question, I have to say that it seems to be more of an Earth sciences question than an Astronomy question. (which would make it off-topic) –  Donald.McLean Jan 12 at 6:53
Though the question as phrased in the body seems too Earth-centric, the question as phrased in the title does seem on-topic: it significantly involves another astronomical body besides Earth. I would suggest rephrasing the body to focus on the rogue planet atmosphere stripping aspect. –  called2voyage Jan 13 at 16:09
Since the OP hasn't come back in a while, I took the liberty of reframing the question around the part relevant to the Astronomy site. –  called2voyage Jan 15 at 15:52

Very interesting question. I'm okay with this question being here since you specifically ask for the effect of rogue planets on Earth's atmosphere. Thinking "astronomically" could be useful in coming to conclusions about whether or not your idea is even a possibility.

When I saw this I instantly thought of ice cores, since trapped gas bubbles could tell you about the composition of the atmosphere. Unfortunately ice cores go nowhere near that far back to be able to tell you about the composition of the atmosphere during the Mesozoic.

Now for the answer:

I can't really speak to the title of your question effectively, since I am unaware of any research involving rogue planets and atmospheric stripping, however, from an astronomy point of view, here is one other thing I would like to point out that may work to reduce the thickness of the atmosphere with time.

• A small fraction of the gasses of the atmosphere exceed the escape velocity of Earth at any given time. Over time, many of these gasses get bombarded by high energy photons (as well as other particles) and fly out of Earth's gravitational potential well. If there are no other sources of these lighter gasses then there is a net loss.

Temperature does play a key role here (as the above plot suggests), but regardless, this process works to reduce the amount of atoms and molecules in the atmosphere (and it affects smaller mass atoms/molecules more so than heavier ones - this is why the Martian atmosphere is primarily carbon dioxide. Essentially it wasn't massive enough to be able to hold on to its lighter gasses, and over time the vast majority of them escaped).

If you had temperature information going back as far as the Mesozoic era, you could in theory run the clock backwards by running a simulation of the evolution of the atmosphere. I don't know how helpful this would really be (or how accurate), since there are potentially other things like volcanic outgassing and the bi-products of living things to complicate the situation. Quite honestly, I know little about these topics and so I couldn't tell you whether there is a net loss or gain of atmospheric gasses.

• As far as rogue planets are concerned, I would find it rather unlikely that such a massive object would still be lurking around the solar system between about 252 to 66 million years ago. The early solar system was certainly more chaotic than it is now (for example, Giant impact hypothesis: $\sim 4.5$ Gya; the collision which caused Uranus' tilt: a few billion years ago), however it is not out of the realm of possibility.
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Why would the rogue planet be 'lurking around the solar system'? Surely it would be hammering past, rip off Earth's atmosphere and disappear into deep space again. Remember that these planets are likely to be gas giants much larger than Jupiter and if they were travelling fast enough, they would not stay long within the clutches of the Sun's gravitational pull. –  user782 Jan 28 at 17:51
And @user782, a large gas giant barreling through the solar system wouldn't affect anything else? –  HDE 226868 Aug 13 at 19:08


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# Arranging letters to make a word in a regular language

Fix a regular language $$L$$ on the alphabet $$\{a, b\}$$, and consider the following problem. I am given as input:

• some number $$m \in \mathbb{N}$$ of copies of the letter $$a$$, and
• some number $$n \in \mathbb{N}$$ of copies of the letter $$b$$ but each copy $$1 \leq i \leq n$$ comes with a constraint expressed as a pair of integers $$(p_i, q_i)$$ which means: "there must be at least $$p_i$$ $$a$$'s to the left of this $$b$$ and at least $$q_i$$ $$a$$'s to the right of this $$b$$".

My goal is to decide if I can construct a word of length $$m + n$$ with $$m$$ letters $$a$$ and $$n$$ letters $$b$$ that falls in the language $$L$$ and where every copy of $$b$$ was put at a position that satisfies its constraints. (Formally: there is an injective function $$f$$ from $$\{1, \ldots, n\}$$ to $$\{1, \ldots, n+m\}$$ such that, letting $$A$$ be the elements of $$\{1, \ldots, n+m\}$$ that are not in the image of $$f$$, for each $$1 \leq i \leq n$$, the set $$A$$ contains at least $$p_i$$ integers that are $$< f(i)$$ and at least $$q_i$$ integers that are $$> f(i)$$.) Note that the $$b$$'s can be put in any order (as long as their constraints are satisfied), they needn't be put in the order in which they are in the input. In other words, $$f$$ need not be an increasing function.

Is this problem in polynomial time for every regular language $$L$$, or is there a language $$L$$ for which the problem is NP-hard?

I have a PTIME greedy algorithm that works if $$L$$ has only one word of every length, e.g., it is something like $$(ab)^*$$. In this case, you should go over the even positions (where $$b$$'s must go) and at each position put a copy of $$b$$ which is available and which is as constrained as possible, i.e., the constraint $$p_i$$ is satisfied, and the constraint $$q_i$$ is as large as possible. It is clear that this algorithm respects the constraints if it succeeds, and that this is always the best way to place the $$b$$'s. However when $$L$$ contains multiple words of length $$m+n$$ then this no longer works and I can't see a dynamic algorithm to solve the problem.

(This question relates to this earlier question of mine but the main difference that the alphabet is now restricted to be $$\{a, b\}$$ so the previous proof does not work. The problem can also be equivalently phrased in terms of topological sorts in which case it is a rephrasing of a problem in this list.)


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Different definitions of the dimension of an algebra

I know of three ways to define the dimension of a finitely-generated commutative algebra A over a field F:

• The Gelfand-Kirillov (GK) dimension, based on the growth of the Hilbert function.
• The Krull dimension, based on chains of prime ideals.
• The transcendence degree of the fraction field of A over F.

According to Artin, GK dimension is the most robust notion because it applies to certain noncommutative algebras. And in the noncommutative setting one can't form fraction fields, so the transcendence degree is out of the question (right?).

But Krull dimension has the advantage that it applies to arbitrary rings. As far as I can tell, the definition of GK dimension only applies to algebras. So: as a matter of pedagogy, which notion of dimension is most appropriate for what applications? Which is easiest to prove things about when?

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In a non-commutative ring, you need to be careful with what you even mean by a prime ideal, and usually there are very few two-sided ideals you might call prime. Oh, and even in the cases when there is a nice ring of fractions, it won't be a field, and so transcedence degree is still bad.

My personal favorite notion of dimension is 'global dimension', the maximum projective dimension of any module of the ring. This concept exists for any ring, and in fact for any abelian category (though, if there aren't enough projectives, you need to play with the definition). The only problem is that it can often be infinity, even for relatively mild rings, like C[x]/x^2. It still makes for a pretty good theory of 'smooth dimension', however.

From a conceptual perspective, Krull dimension seems best suited for geometric perspectives, since it is measuring chains of irreducible closed subsets. The easiest times to work with Krull dimension is when you are in a Cohen-Macaulay ring, and then Krull dimension is equivalent to depth, which is easier to prove things about, since you only need to produce a maximal regular sequence.

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In non-commutative algebra Krull dimension has been generalised by Gabriel & Rentschler. A decent account of it can be found in Chapter 6 of McConnell and Robson's book on non-commutative Noetherian rings.

The basic idea is as follows: An artinian module has Krull dimension 0.

A module that does not have Krull dimension 0 has Krull dimension 1 if in every infinite descending chain of submodules all but finitely many composition factors have Krull dimension 0.

A module that does not have Krull dimension 0 or 1 has Krull dimension 2 if in every infinite descending chain of submodules all but finitely many composition factors have Krull dimension 0 or 1.

The definition continues for all finite ordinals (and can be extended to all ordinals). Then the Krull dimension of a ring R is the Krull dimension of R as a module over itself.

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I perhaps should have said, the more obvious generalisation of Krull dimension using prime ideals is not very good in general because "large" non-commutative rings can have very few prime ideals. – Simon Wadsley Oct 26 '09 at 9:07

Often the most useful dimension in non-commutative algebra is the length of the minimal injective resolution of the ring as a module over itself. In many important cases this is the same as the global dimension when the latter is finite, but it is more robust in that it is finite more often. In a commutative Noetherian ring it is the same as the Krull dimension when it is finite.

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In other words, the injective dimension if the ring as a module over itself. I think the name Gorenstein dimension has more or less stabilized to name this, no? – Mariano Suárez-Alvarez Jan 13 '13 at 0:11
I see 'self injective dimension' more often, I think. But yes, it is just the injective dimension of the ring as a module over itself. – Simon Wadsley Jan 13 '13 at 12:40

I agree with Greg Muller that homological dimension is very nice. In fact various flavours of it turn out to be the same as Krull dimension. For instance a local commutative ring with unit is regular if and only if it has finite global dimension and in this case the global dimension is precisely the Krull dimension.

If there are singularities present so that the global dimension is infinite one can then turn to relative homological dimensions. For instance the right global Gorenstein injective dimension of an n-Gorenstein (not necessarily commutative) ring is n and in fact one can detect Cohen-Macaulayness using a criterion involving Gorenstein injective dimension. One can extend these notions to (at least certain) abelian categories.

To actually try and answer some of your question now - at least in commutative algebra I agree with Greg again that (locally) depth tends to be one of the most useful interpretations of the dimension in the cases (Cohen-Macaulay rings) when it agrees with dimension. Although as I sort of alluded to above one can prove results using homological notions of dimension as well. I don't really know the non-commutative theory well enough to comment in that case.

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I know there has been no action on this question in awhile, but I was surprised no one mentioned weak dimension and I wanted to weigh in. I'm not trying to take anything away from right global dimension (it's my favorite too), but weak dimension seems to work in just as much generality as global dimension and may rule out some pathologies. For example, the right weak dimension (looking at flat dimension of right $R$-modules) is equal to the left weak dimension. Furthermore, weak dimension really does give different information from the notions of dimension discussed so far (for $R$ non-Noetherian). For example, rings of weak dimension zero are exactly Von Neumann Regular rings (these are also the rings for which all prime ideals are maximal).

An article you might enjoy which computes all the dimensions discussed so far for a particular ring is at this link: http://arxiv.org/abs/0912.0723

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There's actually a Gelfand-Kirillov transcendence degree for a noncommutative algebra over a field that generalizes the classical transcendance degree. Gelfand and Kirillov introduced it to prove that the field of fractions for the $n$th Weyl algebra is different for different $n$. Here's a paper by James Zhang that gives the definition and calculates it for several examples.

There's actually a pretty nice theory for when you can form a field of fractions. A Noetherian domain always has a field of fractions, in a construction that closely resembles the commutative case. (This is a special case of Goldie's Theorem.) For a non-Noetherian domains, things are more complicated: the domain can fail to have a field of fractions, and when it does, it won't usually have a unique smallest field it embeds into (unlike the commutative or noncommutative Noetherian case).

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# Are there fermionic bound states that are gauge neutral to all gauge forces?

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Puzzle: In our standard model particle physics, are there fermionic bound states that are gauge neutral to all gauge forces?

Here we concern strong SU(3) color, electromagnetic U(1) EM, hyper U(1), and weak SU(2) forces.

It looks that a single neutron (with 3 quarks $$uud$$) is a bound state, neutral under electric force and strong force. But from the post Are Neutrons gauge neutral to all gauge forces?

• the neutron has a magnetic moment so interacted with magnetic field. (This is a bit strange to me, since we can boost the Lorentz force, which can mix electric and magnetic field. So after boosting the reference frame, the electric neutral neutron can also be electric non-neutral interacting with electric field???)

• the neutron also carries weak charge (how much?)

Are there possible candidates of fermionic bound states that are gauge neutral to all gauge forces?

This post imported from StackExchange Physics at 2020-11-28 23:02 (UTC), posted by SE-user annie marie heart
Does the right handed neutron as a bound state count as gauge neutral to all gauge forces?? $u_Rd_Rd_R$, suppose that the Higgs condensate is zero (so right handed [SU(2) singlet] quarks cannot be paired to the left handed [SU(2) doublet] via mass term.]
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# Halin graph

A Halin graph.

In graph theory, a Halin graph is a type of planar graph. It is constructed from a tree that has at least four vertices, none of which have exactly two neighbors. The tree is drawn in the plane so none of its edges cross; then edges are added that connect all its leaves into a cycle.[1] Halin graphs are named after German mathematician Rudolf Halin, who studied them in 1971,[2] but the cubic Halin graphs had already been studied over a century earlier by Kirkman.[3]

## Construction

A triangular prism, constructed as a Halin graph from a six-vertex tree

A Halin graph is constructed as follows. Let ${\displaystyle T}$ be a tree with more than three vertices, such that no vertex of ${\displaystyle T}$ has degree two (that is, no vertex has exactly two neighbors), embedded in the plane. Then a Halin graph is constructed by adding to ${\displaystyle T}$ a cycle through each of its leaves, such that the augmented graph remains planar.

## Examples

A star is a tree with exactly one internal vertex. Applying the Halin graph construction to a star produces a wheel graph, the graph of a pyramid. The graph of a triangular prism is also a Halin graph: it can be drawn so that one of its rectangular faces is the exterior cycle, and the remaining edges form a tree with four leaves, two interior vertices, and five edges.

The Frucht graph, one of the two smallest cubic graphs with no nontrivial graph automorphisms, is also a Halin graph.

## Properties

Every Halin graph is 3-connected, meaning that it is not possible to delete two vertices from it and disconnect the remaining vertices. It is edge-minimal 3-connected, meaning that if any one of its edges is removed, the remaining graph will no longer be 3-connected.[1] By Steinitz's theorem, as a 3-connected planar graph, it can be represented as the set of vertices and edges of a convex polyhedron; that is, it is a polyhedral graph. And, as with every polyhedral graph, its planar embedding is unique up to the choice of which of its faces is to be the outer face.[1]

Every Halin graph is a Hamiltonian graph, and every edge of the graph belongs to a Hamiltonian cycle. Moreover, any Halin graph remains Hamiltonian after deletion of any vertex.[4] Because every tree without vertices of degree 2 contains two leaves that share the same parent, every Halin graph contains a triangle. In particular, it is not possible for a Halin graph to be a triangle-free graph nor a bipartite graph. More strongly, every Halin graph is almost pancyclic, in the sense that it has cycles of all lengths from 3 to n with the possible exception of a single even length. Moreover, any Halin graph remains almost pancyclic if a single edge is contracted, and every Halin graph without interior vertices of degree three is pancyclic.[5]

Every Halin graph has treewidth at most three.[6] Therefore, many graph optimization problems that are NP-complete for arbitrary planar graphs, such as finding a maximum independent set, may be solved in linear time on Halin graphs using dynamic programming.[7]

The weak dual of an embedded planar graph has vertices corresponding to bounded faces of the planar graph, and edges corresponding to adjacent faces. The weak dual of a Halin graph is always biconnected and outerplanar. This property may be used to characterize the Halin graphs: an embedded planar graph is a Halin graph, with the leaf cycle of the Halin graph as the outer face of the embedding, if and only if its weak dual is biconnected and outerplanar.[8]

## History

In 1971, Halin introduced the Halin graphs as a class of minimally 3-vertex-connected graphs: for every edge in the graph, the removal of that edge reduces the connectivity of the graph.[2] These graphs gained in significance with the discovery that many algorithmic problems that were computationally infeasible for arbitrary planar graphs could be solved efficiently on them,[4][8] a fact that was later explained to be a consequence of their low treewidth.[6][7]

Prior to Halin's work on these graphs, graph enumeration problems concerning the cubic Halin graphs were studied in 1856 by Thomas Kirkman[3] and in 1965 by Hans Rademacher.[9] Rademacher calls these graphs based polyhedra. He defines them as the cubic polyhedral graphs with f faces in which one of the faces has f − 1 sides. The graphs that fit this definition are exactly the cubic Halin graphs.

The Halin graphs are sometimes also called roofless polyhedra,[4] but, like "based polyhedra", this name may also refer to the cubic Halin graphs.[10] The convex polyhedra whose graphs are Halin graphs have also been called domes.[11]

## References

1. Encyclopaedia of Mathematics, first Supplementary volume, 1988, ISBN 0-7923-4709-9, p. 281, article "Halin Graph", and references therein.
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# An application of Fubini's theorem

Let $v_n$ be the volume of the unit ball in $\mathbb{R}^n$. Show by using Fubini's theorem that $$v_n = 2\,v_{n-1}\,\int_0^1{(1-t^2)^{(n-1)/2}\,dt}\,.$$

I've been trying to do this inductively, and I've been thinking about the integral as kind of a generalization of the integral formula for the area of (part of) the unit disc in $\mathbb{R}^2$, but when I go into more dimensions, the exponent is tripping me up.

One of my thoughts was to just write down an integral for the volume of the n-ball as a big iterated integral, $$4\,\int_0^1{\cdots \int_0^1{(1-x_1^2)^{1/2}\cdots (1-x_n^2)^{1/2}\,dx_n\cdots dx_1}}\,,$$ but the more I look at this, the less sense it makes to me. At that point, I was thinking of, like, isolating one coordinate at a time and integrating, where the other coordinates are "smooshed" together in a "y-axis." Or something.

Edit: Here's a solution (I believe).

In general, we have $$v_n = \idotsint\limits_{\sum\limits_{i=1}^n{x_i^2}\le 1}{dx_1\cdots dx_n}\,.$$ So by Fubini's theorem, we have $$\int_{-1}^1{\left[\idotsint\limits_{\sum\limits_{i=1}^{n-1}{x_i^2}\le 1-x_n^2}{dx_1\cdots dx_{n-1}}\right]\,dx_n}\,.$$

Now set $y_i=\dfrac{x_i}{\sqrt{1-x_n^2}}$. So $dy_i=\dfrac{dx_i}{\sqrt{1-x_n^2}}$. Thus, we have $\sum\limits_{i=1}^{n-1}{y_i^2} = \sum\limits_{i=1}^{n-1}{\dfrac{x_i^2}{1-x_n^2}}\le 1$.

So we make our change of variables: $$\int_{-1}^1{\left[\idotsint\limits_{\sum\limits_{i=1}^{n-1}{y_i^2}\le 1}{\left(\sqrt{1-x_n^2}\right)^{n-1}\,dy_1\cdots dy_{n-1}}\right]\,dx_n} = 2\,\int_0^1{\left(\sqrt{1-x_n^2}\right)^{n-1}v_{n-1}\,dx_n}\,.$$ Now we can just call $x_n$ whatever we like -- in this case, $t$.

How does this look?

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You are almost there. You should note that the leading constant factor in your expression for the volume of the $n$-ball should not be 4, but $2^n$. –  Willie Wong Jan 25 '11 at 0:15
@Willie: This clears up some of my trouble, but I still can't get the exponent to work out correctly... –  Bey Jan 25 '11 at 1:46
Your integrand is also a little faulty. (This has to do with what Theo wrote below.) The ball is represented by $\sum x_i^2 \leq 1$, but your integrand admits points with coordinate $(1,1,1,\ldots,1)$ which is not in the sphere. –  Willie Wong Jan 25 '11 at 12:32
@Willie: I figured it out, I believe. I'll be editing my post to include the solution. –  Bey Jan 25 '11 at 16:38
looks good. The change of variable trick you used is similar to the scaling argument Theo described. Good work. –  Willie Wong Jan 25 '11 at 17:04
Hint: How does the $(n-1)$-dimensional volume of a ball scale with its radius $r$?
@Bey: Slice the ball with horizontal planes orthogonal to the $t$-axis at height $t$. Then $r = (1-t^{2})^{1/2}$ is the radius of the slice of the ball at height $t$. Now observe that $v_{n-1}(r) = r^{n-1}v_{n-1}(1)$, where $v_{n-1}(r)$ is the volume of the $(n-1)$-dimensional ball of radius $r$. –  t.b. Jan 25 '11 at 1:19


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• # question_answer Study the following passage carefully and answer the questions given below: There was a spike in farmer suicides in the end of the green revolution. Small scale farmers were exposed to price fluctuations that are characteristic of cash crops on the global market and were in competition with multinational corporations. Large scale Indian farmers adopted the use of pesticides, fertilized and genetic modified seeds. It raised the production costs significantly. The small scale farmers had to take out loans for inputs because they were expensive and couldnt afford them. Why was small scale farmers involved in suicides? A) Green revolution spiked them to expose global market B) They were crushed under international competition C) They were poor farmers to purchase pesticides D) Cash crops are not entertained in global market

 Rationale: (b) Reasoning type: Cause-and-Effect Reasoning$\to$showing causes and resulting effect Focus of the question: Conclusion$\to$Finding the focal point of the passage Relationship analogy: Cause and Effect Analogy$\to$It is a relationship between events or things, where one is the result of the other or others The question focuses on the conclusion of the passage with respect to causes of farmers suicides. The small scale farmers were exposed to price fluctuations that are characteristic of cash crops on the global market and were in competition with multinational corporations.


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# Extracting specific features using HOG

I am using HOG (Histogram of Oriented Gradients) for car detection from a video. I have used the Matlab function extractHOGFeatures() , it has given me a feature vector but how do I differentiate between different features of different objects. After this I extracted the corner points(313x1) and BRISK points (172x1) but they show nothing just empty vectors.