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Chapter 61
###### Figure 61–1.

Structure of the skin.

### Epidermis

The epidermis is composed of stratified squamous epithelium in which several cell types can be identified.

#### Keratinocytes

Keratinocytes are subdivided according to degree of differentiation into the following four types: (1) basal cells, the germinative, epidermal stem cells, which are fixed to the basement membrane by attachment plaques (modified desmosomes) and anchoring filaments (stratum basale); (2) prickle or spinous cells (stratum spinosum), characterized by the presence of keratin and desmosomes and named for the desmosomal intercellular attachments, which appear as spines or prickles connecting cells; (3) granular cells (stratum granulosum), containing numerous large basophilic keratohyalin granules in the cytoplasm; and (4) cornified cells (stratum corneum)—flat, anucleate cells that form the most superficial layer of the epidermis. The stratum corneum is continuously shed at the surface and varies greatly in thickness—it is greatest in the soles and palms.

The normal rate of maturation from a basal cell to a surface cornified cell is 2 weeks. The cell remains in the stratum corneum another 2 weeks before it is shed.

#### Melanocytes

Melanocytes are epidermal cells derived from the embryologic neural crest. They produce melanin and are situated singly in the basal layer, appearing as larger clear cells. The dopa reaction, which is positive in the presence of enzymes of the tyrosine–melanin pathway is a histochemical means of identifying melanocytes. Melanocytes have dendritic processes that ramify in the epidermis and transfer melanin to keratinocytes.

Melanocytes can be identified through electron microscopy by the presence of melanosomes, which are membrane-bound ellipsoidal structures containing concentric internal lamellae. Positive staining for S100 protein and melanosomal antigen (HMB45) are useful immunohistochemical markers for melanocytes.

The number of melanocytes in the skin is relatively constant. Skin pigmentation is dependent on the rate of synthesis of melanin, which is governed by (1) racial factors, being greater in dark-skinned races; (2) ultraviolet radiation, which increases melanin synthesis; and (3) hormones—melanocyte-stimulating hormone and adrenocorticotropin both increase melanin pigmentation.

#### Langerhans Cells

Langerhans cells are clear dendritic cells situated among the cells of the stratum spinosum. They are believed to be antigen-processing cells. They are S100 protein-positive on immunohistochemical studies. On electron microscopy, they lack melanosomes but contain a characteristic organelle known as a Birbeck granule.

#### Merkel Cells

Merkel cells are neuroendocrine cells that are present in the basal layer of the epidermis. They cannot be recognized in routine histologic sections but can be identified in electron micrographs by the presence of cytoplasmic neurosecretory granules.

### The Dermis

The dermis is a layer of connective tissue 1–4 mm thick subjacent to the epidermis. ...

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text:
# Prove that $\cos(x+y)=\cos(x)\cos(y)-\sin(x)\sin(y)$ using Cauchy product.

Show the correctness of the equality by making use of the Cauchy product of the series $\cos(x)=\sum_{n=0}^{\infty}(-1)^n.\frac{x^{2n}}{(2n)!}$ and $\sin(x)=\sum_{n=0}^{\infty}(-1)^n.\frac{x^{2n+1}}{(2n+1)!}$. I've reached $$\cos(x).\cos(y)=\sum_{n=0}^{\infty}(-1)^n.\sum_{k=0}^{n}\frac{1}{(2k)!(2(n-k))!}.x^{2k}.y^{2(n-k)}$$ and $$\sin(x).\sin(y)=\sum_{n=0}^{\infty}(-1)^n.\sum_{k=0}^{n}\frac{1}{(2k+1)!(2(n-k)+1)!}.x^{2k+1}.y^{2(n-k)+1}$$ and $$\cos(x+y)=\sum_{n=0}^{\infty}(-1)^n.\sum_{k=0}^{2n}\frac{1}{k!(2n-k)!}.x^{k}.y^{2n-k}$$

but I guess I'm missing some steps or got something wrong.

Everything is fine so far. We can start from \begin{align*} \cos(x)\cos(y)&=\sum_{n=0}^\infty (-1)^n\sum_{k=0}^n\frac{x^{2k}y^{2n-2k}}{(2k)!(2n-2k)!}\tag{1}\\ \sin(x)\sin(y)&=\sum_{n=0}^\infty (-1)^n\sum_{k=0}^n\frac{x^{2k+1}y^{2n-2k+1}}{(2k+1)!(2n-2k+1)!}\tag{2}\\ \cos(x+y)&=\sum_{n=0}^\infty (-1)^n\sum_{k=0}^{2n}\frac{x^{k}y^{2n-k}}{k!(2n-k)!}\\ \end{align*}

We obtain \begin{align*} \cos(x+y)&=\sum_{n=0}^\infty (-1)^n\sum_{k=0}^{2n}\frac{x^{k}y^{2n-k}}{k!(2n-k)!}\tag{3}\\ &=\sum_{n=0}^\infty (-1)^n\sum_{k=0}^{n}\frac{x^{2k}y^{2n-2k}}{(2k)!(2n-2k)!}\\ &\qquad\qquad +\sum_{n=1}^\infty (-1)^n\sum_{k=0}^{n-1}\frac{x^{2k+1}y^{2n-2k-1}}{(2k+1)!(2n-2k-1)!}\tag{4}\\ &=\cos(x)\cos(y)+\sum_{n=0}^\infty (-1)^{n+1}\sum_{k=0}^{n}\frac{x^{2k+1}y^{2n-2k+1}}{(2k+1)!(2n-2k+1)!}\tag{5}\\ &=\cos(x)\cos(y)-\sin(x)\sin(y) \end{align*} and the claim follows.

Comment:

• In (3) we split the inner sum in even and odd parts. We do so by replacing $k\rightarrow 2k$ and $k\rightarrow 2k+1$.

• In (4) we see the left-hand series is already $\cos(x)\cos(y)$ according to (1). We shift the index $n$ of the right-hand series by one to start from $n=0$.

• In (5) we see the right-hand series is $-\sin(x)\sin(y)$ according to (2).

• The splitting trick is quite useful! I just want to point out a typo in (4) and (5): the factorial in the denominator should be $(2k+1)!$, not $(2k-1)!$. Dec 7 '16 at 17:37
• @HaidarAbboud: Typo corrected. Thanks for the hint! Dec 7 '16 at 17:44

format: plain


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# Elastic net regularized regression for time-series analysis of plasma metabolome stability under sub-optimal freezing condition

### Subjects

An Author Correction to this article was published on 15 June 2018

## Abstract

In this paper, the stability of the plasma metabolome at −20 °C for up to 30 days was evaluated using liquid chromatography-high resolution mass spectrometric metabolomics analysis. To follow the time-series deterioration of the plasma metabolome, the use of an elastic net regularized regression model for the prediction of storage time at −20 °C based on the plasma metabolomic profile, and the selection and ranking of metabolites with high temporal changes was demonstrated using the glmnet package in R. Out of 1229 (positive mode) and 1483 (negative mode) metabolite features, the elastic net model extracted 32 metabolites of interest in both positive and negative modes. L-gamma-glutamyl-L-(iso)leucine (tentative identification) was found to have the highest time-dependent change and significantly increased proportionally to the storage time of plasma at −20 °C (R2 = 0.6378 [positive mode], R2 = 0.7893 [negative mode], p-value < 0.00001). Based on the temporal profiles of the extracted metabolites by the model, results show only minimal deterioration of the plasma metabolome at −20 °C up to 1 month. However, majority of the changes appeared at around 12–15 days of storage. This allows scientists to better plan logistics and storage strategies for samples obtained from low-resource settings, where −80 °C storage is not guaranteed.

## Introduction

The importance of pre-analytical processes on blood samples used for subsequent metabolomics analyses has been thoroughly investigated in the past1. Strict observance of pre-analytical protocols, especially storage of samples, is necessary since highly dynamic and pronounced changes in the metabolome occurs as soon as the blood samples are drawn from the patients and continue as the samples are being prepared for metabolomics analysis. The instability of the plasma metabolome under different storage conditions has been previously reported, and there is a general consensus that an acceptable storage temperature for these biological samples should be −80 °C, especially for long-term storage1,2,3. For short-term storage, several reports agree that storage of plasma at −20 °C for one week does not cause significant deterioration of the plasma metabolome2,3.

This ultralow temperature requirement however presents a problem when the source of the biological material is a low-resource area, where a certified −80 °C storage and logistics cannot be guaranteed. Hence, as metabolomics is now being used as a powerful tool to investigate neglected tropical diseases4, it is useful to follow the slow deterioration of the plasma metabolome under sub-optimal freezing conditions, such as at −20 °C, in order to have a better estimate of how long these biological materials can be stored under limited resources for subsequent metabolomics analysis in a more equipped laboratory. This information will help logisticians to plan a better product flow strategy that is a good compromise between sample quality and resource limitations.

Data analysis of plasma metabolome stability in a time series often involves univariate analysis to determine the number of significantly different metabolites that changed across the entire time series. However, as this is univariate, interaction of 2 or more metabolites, that may not be statistically different individually, will not be determined. For time-series data, several methods have been proposed such as COVAIN toolbox (implemented in MATLAB©)5, partial least squares discriminant analysis (PLS-DA)6, and multivariate empirical Bayes statistical time-series analysis (MEBA)7. MEBA for instance, has been increasingly used as it is now integrated in the web-based easy-to-use platform, MetaboAnalyst8,9. MEBA is a time-course analysis method based on multivariate empirical Bayes statistic which could evaluate the importance of the temporal changes of metabolites using the Hotelling’s T2. The outcome is a ranked list of all metabolites that show differences in their temporal profile7. Metabolites with higher Hotelling’s T2 value comprise those whose profiles are more different across the time series. As this method ranks all metabolites, it is difficult to determine which Hotelling’s T2 value indicates that the particular metabolite is substantially changing from those whose temporal change is negligible.

In this paper, we describe a computational approach for the prediction of storage time of plasma at a sub-optimal freezing condition (−20 °C) based on the metabolome profile, which simultaneously ranks metabolites based on their model contribution while removing metabolites whose temporal changes are negligible through the entire time series using elastic net regularized regression. Elastic net is a generalization of the ridge regression and least absolute shrinkage and selection operator (LASSO)10. The combination of ridge and LASSO performs feature selection and handles multicollinearity within the dataset11, which are important characteristics for analysing datasets with large numbers of features (many of which could be collinear) and relatively smaller number of observations typical of omics studies12.

## Results

### Data pre-processing

LC-MS data were pre-processed using Progenesis QI. The processing resulted to the deconvolution and integration of 6749 and 5168 features in positive and negative modes, respectively. Progenesis QI uses both isotope and adducts deconvolution algorithm that identifies the parent ion and integrates the detected adducts (Na, K) into one feature. After filtering of the features based on QC dilutions correlation, %CV in QC samples and number of missing values, the final number of features amounted to 1229 and 1483 for positive and negative modes, respectively. These filtered datasets were then imported to MetaboAnalyst for further data processing and analysis.

### Univariate analysis

One-way repeated measures ANOVA revealed that 10 and 2 features significantly changed throughout the entire time series in positive mode and negative modes, respectively (Table 1). These features comprised of both water soluble and hydrophobic metabolites as evinced by their retention times through an HSS T3 C18 column, which was selected as it affords the retention of both hydrophilic and hydrophobic compounds13.

By using a pair-wise comparison, the number of significantly different metabolite features in all time points were relatively low compared to day 0, with the maximum number of 13/1229 significantly different features appearing at day 18 in positive mode and 12/1483 significantly different features appearing at day 21 in negative mode (Fig. 1). Overall, the data suggests that plasma is relatively stable at −20 °C up to a month with very minimal degradation. A spiked increase in significantly different features at day 18 (positive mode) and 21 (negative mode) is curious and the reason is unknown. However, a similar pattern was reported for the metabolome stability of urine at −20 °C14.

### Multivariate empirical Bayes ANOVA (MEBA)

To determine the features/metabolites whose profile changed throughout the time series, data was first analyzed using the multivariate empirical Bayes ANOVA7 (MEBA) implemented in MetaboAnalyst8,15.

As shown in Fig. 2, highly ranked metabolites in terms of Hotelling’s T2 statistics also showed significant differences over the entire time-course using repeated measures ANOVA. Considering that MEBA is in fact an extension of ANOVA, this result is therefore expected. The metabolite with a neutral mass of 260.1360 Da ranked highest in both positive and negative modes.

### Elastic net regularized regression model

As MEBA gives a rank to all metabolites detected, it is difficult to estimate which Hotelling’s T2 value indicates that the metabolites are actively changing through the time series. Hence, in this paper, we propose a method which ranks metabolites according to their temporal profile and also eliminates metabolites whose temporal change are not large enough to affect the metabolome significantly. The main assumption is that if the plasma metabolome changes during storage at −20 °C in a particular fashion, it is possible to predict the number of days that the plasma has been stored at −20 °C based on the metabolome profile. In this case, it is expected that a deteriorating metabolite would gradually decrease over time and a breakdown product is expected to gradually increase over time. Thus, an elastic net regularized regression model, which is a combination of the ridge regression and LASSO, was employed. Using this method, a feature selection strategy has been implemented that provides a list of metabolites that are predictive of storage time and ranks them based on their contribution to the model and consequently based on their temporal change.

To build the model, a range of α values (0, 0.25, 0.5, 0.75, 1) was initially screened in order to select the best model, which could yield the lowest error. In the end, α = 0.5 was chosen, which gives equal weight to ridge regression and LASSO. In this case, we benefit both from the feature selection capability of LASSO and the ability of ridge regression to handle multicollinearity in the dataset. The optimum λ parameter on the other hand was determined based on cross-validated MSE. Figure 3 shows the metabolites extracted by the elastic net model along with their coefficients. Metabolites with higher absolute value of their coefficients signify higher contribution to the model predicting storage time. From 1229 metabolites detected in positive mode and 1483 metabolites in negative mode, the elastic net model was able to reduce the dataset into 32 metabolites in both positive and negative modes. The metabolites extracted by the elastic net model also ranked relatively high in MEBA, although not all of the top ranked metabolites of MEBA were extracted by the elastic net model. These metabolites are those with erratic changes in time and whose increase nor decrease does not significantly correlate with storage time. For example, metabolite feature 0.31_414.8628 (rt_m/z) ranked 10th based on MEBA and was found to be statistically different over time based on ANOVA (FDR p-value = 0.0460). However, as its temporal change is erratic do not seem to follow a certain increasing or decreasing pattern (R2 = 0.0001, p = 0.91), metabolites as such may not be considered as a biomarker to assess or predict plasma degradation during storage at sub-optimal freezing conditions.

Strong significant correlations (Fig. 4) were found between the actual storage times and predicted storage times in both positive and negative ionization modes, and both training and test sets. This indicates that the models are highly predictive. However, considering that elastic net imposes a constraint to the coefficients, no statistical significance (p-value) can be provided since the standards errors cannot be calculated from biased estimators. Instead of calculating the standard errors for biased coefficients, bootstrapping was performed during cross validation to assess the reproducibility of the results16. Several iterations of the model with 10-fold cross validation yielded the same metabolite features, which indicate the robustness of the selected metabolites for the prediction of storage time at −20 °C conditions.

Focusing on these extracted metabolites, a heat map shown in Fig. 5 graphically represents the degradation pattern of the plasma metabolome across the time series (30 days) at −20 °C. Using principal components analysis (PCA) for data visualization, it can be seen that there is a clear separation between samples stored from 0–12 days and those stored from 15–30 days in positive mode. In negative mode however, there is no clear-cut separation based on storage time. Nonetheless, the heat map (for negative mode) suggests that, similar to positive mode, more drastic changes in metabolite averaged intensities are observed at around 12–15 days of storage.

Based on the extracted metabolites, the most contributing metabolite feature in both positive and negative modes was m/z 261.1414 [M + H]+ or 260.1360 (neutral mass after adduct deconvolution). This metabolite was tentatively identified as L-gamma-glutamyl-L-(iso)leucine. The intensity of this metabolite was found to be significantly correlated with storage time (R2 = 0.6378 [positive mode], R2 = 0.7893 [negative mode], p-value < 0.00001) (Fig. 6). This metabolite was also the highest ranked based on Hotelling’s T2 in MEBA in both positive and negative modes and was found to significantly change over time based on ANOVA (p = 1.26 × 10−16 [positive mode], p = 7.78 × 10−25 [negative mode]).

## Discussion

As metabolomics is rapidly expanding, thanks to developments in instrumentation and data analysis tools, several mechanisms previously unknown in certain diseases are now slowly being uncovered. This therefore provides an opportunity for scientists to look back at diseases that are currently being neglected, and yet still affecting millions of lives. Metabolomics, as appropriately titled in a 2015 paper “enter the battle against the neglected tropical diseases”4, is now being used as an important tool in studying severe acute malnutrition in children17,18 and many other diseases in low-resource areas4. However, although one must aim to achieve the highest level of quality, the storage and cold-chain logistical requirements of samples for metabolomics analysis is sometimes limiting the opportunity for this technique to be used in applications where the source of the biological material has limited resources. Unless the study site has a fully equipped laboratory and/storage facility, ensuring optimal freezing conditions of the samples could be daunting.

The effect of storage temperature on the stability of the plasma metabolome has already been described in the literature using different analytical platforms, such as gas chromatography – MS19, capillary electrophoresis – MS20, nuclear magnetic resonance (NMR) spectrometry3, and LC-MS1,2,21. According to these findings, plasma samples suited for metabolomics analyses should be stored at ≤−80 °C for long-term storage due to instability of many of the metabolites in plasma. In fact, the European Prospective Investigation into Cancer and Nutrition (EPIC) consortium in 2005 recommended the storage of plasma in liquid nitrogen for optimal preservation of metabolites, including rapidly degrading metabolites such as vitamin C22. Although liquid nitrogen or dry ice can be procured in many capital cities even in low-resource countries, delivering a constant supply of liquid nitrogen and dry ice cannot always be guaranteed, especially when the study site is very distant from a bigger city. Further, local flights often restrict the transport of these materials on-board. Also, installation of a −80 °C freezer could be a problem in low-resource settings where supply of electricity could be an issue. Therefore it is very interesting to determine whether significant deterioration of the plasma metabolome occurs upon temporary storage under sub-optimal conditions, such as −20 °C which is almost equivalent to a domestic freezer that can be easily maintained. Also, several passive cold chain procedures can sustain a temperature of −20 °C for a few days in case of power interruptions or while awaiting transfer to a −80 °C freezer in a more equipped laboratory.

Several papers have already reported that storage of plasma until 1 week at −20 °C is acceptable as no significant deterioration of the plasma metabolome was detected at this condition2,3. However, storage for up to 1 month at −20 °C causes deterioration of the plasma metabolome which could render it unfit for further metabolomics analyses2,3. While this information is very substantial, the amount of time between 1 week and 1 month could already dictate whether or not it is feasible to collect samples from low-resource areas. It is therefore important for us to estimate how long within the 1 month time frame is plasma still stable and fit for metabolomics analysis. Therefore, we have conducted a more closely timed experiment to assess the storage stability of the plasma metabolome under −20 °C, and to determine which biomarkers could be used to assess the stability of the plasma metabolome under these conditions.

Univariate analysis suggest that more metabolites were significantly different at 18 and 21 days in positive and negative modes, respectively. We however argue that a better analysis of metabolome stability should be to assess the individual change of these metabolites through time, which could not captured by univariate analysis. Thus in this paper, we report a time-resolved analysis of plasma metabolome deterioration under −20 °C using elastic net regularized regression implemented by the glmnet function in R. Compared to MEBA, where all metabolites are ranked based on their Hotelling’s T2 value, elastic net is a penalized regression model that includes a feature selection step, which extracts metabolite features that are predictive of storage time. The glmnet method used herein employed a Poisson regression model, which is a log-linear regression model to predict count data. Such method has been successfully used to predict the number of days a patient is likely to stay in a hospital using data from the electronic patient record system23. Similarly in this paper, we predicted the number of days the plasma has been stored at −20 °C based on the intensities of the metabolites detected by LC-high resolution MS. Glmnet has been employed in several studies in the field of genomics, especially associating gene expression and gene methylation with disease outcomes10,24,25. Glmnet, especially LASSO, has also been used in various metabolomics studies. For instance, this method has been used to predict the acamprosate treatment response in alcohol-dependent subjects based on metabolomics profile and clinical data26, and the age and sex of participants of the Karlsruhe Metabolomics and Nutrition (KarMeN) study based on their metabolomic profile27. To the best of our knowledge, this is the first report using elastic net on a time-series data for metabolomics analysis.

The elastic net model produced a highly predictive model with significantly high correlations between the actual storage time and the prediction in both training and test sets. The most contributing metabolite feature in both positive and negative modes was tentatively identified as L-gamma-glutamyl-L-(iso)leucine based on database comparison to the Human Metabolome Database (HMDB)28. This coincided with the results of MEBA and ANOVA, which supports the validity of the generated models. According to the HMDB, L-gamma-glutamyl-L-(iso)leucine is a proteolytic breakdown product of larger proteins. This metabolite is a plasma/serum biomarker for non-alcoholic fatty liver disease and non-alcoholic steatohepatitis29 and has been reported to increase in plasma of obese patients after consumption of whey isolate30. As this metabolite is a protein breakdown product, it is not surprising that the level of this metabolite is increasing with increasing storage time potentially due to the residual proteolytic activity that is not quenched at −20 °C.

As MEBA ranks all metabolites based on their Hotelling’s T2 value, which is indicative of the degree of their temporal changes, there are no absolute criteria on which Hotelling’s T2 value and/or rank signify that certain metabolites are considered to be of interest. For instance, in a paper studying the change in concentration of a potential biomarker for hepatocarcinoma over time, the top 30 metabolites with the highest Hotelling’s T2 value were considered as the final feature subset31. On the other hand, a study on urine metabolomic analysis to detect metabolites associated with the development of contrast induced nephropathy considered the top 4 metabolites based on Hotelling’s T2 values to be of interest32. Furthermore, a time-resolved metabolomics analysis of individual differences during the early stage of lipopolysaccharide-treated rats extracted 15 metabolites out of 60 identified metabolites as metabolites of interest based on Hotelling’s T2 rank33. In contrary, the proposed method in this paper is able to objectively distinguish which metabolites are changing across the time series and their temporal changes can be ranked based on the absolute value of the model coefficients.

Although we have pointed the advantages of using this approach, the method proposed in this paper applies only to one-sample problems (time-series data without additional independent variables) contrary to MEBA, which can resolve the interaction of time and another independent variable (i.e. treatment)8. Further research is required to develop a regularized regression model for datasets with additional independent variables. Furthermore, MEBA is designed to capture temporal changes even when there is no particular time-course patterns. In this paper, the assumption that degradation would elicit a time-course increase or decrease of certain metabolites was held. Hence, the utility of this model may only be best if a linear temporal pattern is expected.

Based on the extracted metabolite features using elastic net regularized regression, it can be seen in Fig. 5 that the metabolome appears to dramatically shift beginning around 12–15 days. Compared to previous data in the literature which suggests that 1 week storage at −20 °C is acceptable, our results show that plasma metabolome may still be used up to 2 weeks with very minimal changes in the profile. However, it is also noteworthy that the number of metabolites changing within the 1-month time frame are very minimal, which may suggest that plasma may in in fact be relatively stable at −20 °C for up to 1 month. This give more opportunity for field scientists to plan a strategic logistics cold-chain to obtain samples from low-resource areas for subsequent metabolomics analysis.

## Materials and Methods

### Plasma preparation and storage

Plasma samples were obtained from blood donors of the local blood bank (Ghent, Belgium) who have provided a signed informed open consent for the use of their blood for research purposes. Blood samples from 4 healthy donors were obtained in January 2017, immediately processed into plasma and blast frozen at −30 °C on-site and delivered to the lab. Plasma samples were then thawed in ice in the lab for further processing. As this study focused on the storage stability of plasma and not on the inter-individual variability, plasma samples from the four patients were pooled to generate a plasma reference material. This plasma mixture was used in lieu of (and to replicate) the NIST SRM 1950, which is expensive and offered in limited supply. The plasma mixture was then pipetted into 500 µL aliquots in 1.5 mL Eppendorf tubes. This research, including the methods performed herein, were approved by the Ghent University Commission for Medical Ethics (file number: EC/2017/0095) and are in accordance with its guidelines and regulations.

For the storage stability analysis, 100 tubes were immediately frozen at −20 °C while 10 samples were immediately frozen at −80 °C, which served as control (Time 0). Ten tubes were then transferred to −80 °C every 3 days. The long term stability of plasma at −80 °C has been demonstrated and thus it is inferred that the control sample would remain stable during the entire duration of the storage test2. Further, upon transfer of samples from −20 °C to −80 °C, degradation would most likely retard, thus keeping the current state of the metabolome stable until analysis.

### Metabolite extraction

At day 30, all plasma samples were thawed in ice for metabolite extraction. Plasma metabolites were extracted using ice-cold methanol pre-mixed with internal standards. To each 500 µL plasma aliquot, 1 mL of extraction solvent was added followed by rigorous vortexing and storage at −20 °C for an hour to allow protein precipitation. Thereafter, the tubes were centrifuged at 13.000 × g for 10 minutes at 4 °C. The supernatant (750 µL) was transferred to a glass tube and dried using a gentle stream of nitrogen at <10 °C. The dried pellets were reconstituted with 10% acetonitrile containing 0.1% formic acid and subjected to liquid chromatography-mass spectrometry analysis.

An aliquot of 10 µL was taken per sample and added together. This new sample served as a QC sample. A dilution series of the QC sample was made by diluting the QC sample in 0.1% formic acid at 1:1, 1:5; 1:10, and 1:50 (QC sample:water).

### Metabolite profiling

Chromatographic separation was performed using a Waters Acquity UPLC I-class FTN system (Waters, Manchester, UK) using both mobile phase solvent and flow rate gradients as suggested by Barri et al.13 with modifications. Mobile phase and flow rate gradients were adjusted between positive and negative modes to allow better peak separation within each ionization mode, as depicted in Table 2. The mobile phases consisted of 0.1% formic acid (A) and 0.1% formic acid in acetonitrile (B). Peaks were separated using an Acquity UPLC HSS T3, 100 Å, 1.8 µm, 1 mm × 100 mm column. Injection volume was 5 µL and the column was maintained at 50 °C.

The LC flow was directed to a Waters Synapt G2Si high resolution mass spectrometer (Waters, Manchester, UK) via electrospray ionization (ESI) in both positive and negative modes. Ionization capillary voltage was set at 2.75 kV for positive mode and 2.20 for negative mode. Source and desolvation temperatures were 150 °C and 500 °C, respectively. Cone and desolvation gas flow rates were 20 L h−1 and 600 L h−1. Mass range was 50–1000 Da and a scan speed of 0.1 s was applied in MSE centroid resolution mode. MSE collision energy was a ramp of 10–30 V for both positive and negative modes. Prior to mass acquisition, the mass spectrometer was calibrated using a mixture of sodium formate adducts. Internal calibration and online exact mass correction was applied using leucine-enkephalin (200 pg µL−1), which was infused to the MS every 10 seconds during acquisition at a flow rate of 20 µL min−1.

A blank sample (100% acetonitrile) was injected 10 times at the beginning of the analysis to condition the column. Then, the QC sample and dilutions of the QC sample were injected to the LC-MS. Blank and QC sample were re-injected after every 10 experimental samples in the run. Samples were analyzed in a random order.

### Data preprocessing and MEBA analysis

Data acquisition was monitored using MassLynx v1.7 (Waters). After data acquisition, data was imported to Progenesis QI (Nonlinear Dynamics, Newcastle, UK) for data pre-processing, including peak picking and deconvolution, and peak alignment using default settings from retention times 0 to 8.5 minutes. Thereafter, the data was exported to Excel (Microsoft) for data inspection and filtering. First, zero values were replaced with blank (empty cells). Features found in the QC samples that did not respond to dilution (R2 > 0.5; dilution factor versus feature intensities) were considered background noise and thus removed from the dataset. Then, features of the QC samples whose coefficient of variation were greater than 30 (%CV > 30) were also removed. Finally, features from the entire dataset which had more than 20% missing values were removed.

Univariate analysis and MEBA were performed using the time-series analysis tool in MetaboAnalyst8,9. Missing values were imputed using the K-Nearest Neighbor (KNN) algorithm and the data was normalized to constant sum and transformed using the generalized log transformation and autoscaling. One-way repeated measures ANOVA was used to determine features that significantly changed across the time series. Multiple testing correction was done using Benjamini-Hochberg correction34 at false discovery rate (FDR) p < 0.05. MEBA on the other hand was used to rank the features according to the differences in their temporal profiles based on the Hotelling’s T2 value, such that features with higher value are those that have higher time-course changes.

### Elastic net regularized Poisson regression

The pre-processed data was then used to build an elastic net regularized regression model using the glmnet package35 in R36. Elastic net is a generalized linear model that operates as a mix of ridge regression and LASSO, which was specifically designed to overcome issues of large variable number (metabolite features) and small sample size. Poisson regression on the other hand is used to model count data. In this case, we used the Poisson regression to model the number of storage days based on the metabolomic profile of each plasma sample. The log-likelihood function is given by Equation 135:

$$l(\beta |X,Y)=\,\sum _{i=1}^{N}(yi({\beta }_{0}+\beta ^{\prime} {x}_{i})-{e}^{{\beta }_{0}+{\beta }^{T}{x}_{i}}.$$
(1)

where the elastic-net penalty is defined as Equation 2

$$\mathop{{\rm{\min }}}\limits_{{\beta }_{0},\beta }-\frac{1}{N}l(\beta |X,Y)+\lambda ((1-\alpha )\sum _{i=1}^{N}{\beta }_{i}^{2}/2+a\sum _{i=1}^{N}|{\beta }_{i}|\,)$$
(2)

where β0 and β are the coefficients of the linear model and N is the number of samples. The contribution of ridge regression and LASSO to model generation can be adjusted by adjusting the elastic net mixing parameter α, with range α  [0, 1]. α = 1 is the LASSO and α = 0 is for ridge regression. Values in between adjust the contribution of ridge regression and LASSO α = 1 − ε for some ε < 0.5 performs much like LASSO while α = 1 − ε for some ε > 0.5 performs much like ridge regression. α = 0.5 equally imposes LASSO and ridge regression on the model. Before the data was passed to glment, the data were centered to zero and hence glmnet’s standardize option was set to false (standardize = FALSE).

The parameter λ is the tuning parameter (λ ≥ 0) which controls the strength of the shrinkage of the variables (metabolite features) and is optimized by cross validation. A leave-one-out cross-validation was used to determine the optimal value of regularization parameter λ based on both the minimum mean squared error (MSE) and minimum MSE + 1 standard error (SE) of the minimum MSE using the “cv.glmnet” function. The optimal λ value was then used for feature selection and model generation. Since the model aims to predict the number of days the plasma samples are in storage based on the metabolome profile, the data was fit in a Poisson regression model (family = “poisson”) and the loss to use for cross-validation was set to “type.measure = deviance”.

To build the model, the data was initially split into training and test sets using the createDataPartition function in the caret package37. The data was randomly split into training set (75% of the data set) and test set (25%). Model generation was performed using the training set and the test set served as an external validation of the model. The model was assessed by calculating the root mean squared error (RMSE) of the training set and root mean squared error of prediction (RMSEP) of the test set, and the correlation between the number of days the plasma was stored at −20 °C and the predicted storage days using Pearson correlation (R2 > 0.8, p < 0.05) in both training and test sets. Features extracted from the elastic net Poisson model were ranked based on the absolute values of the model coefficients, such that higher values indicate more important contribution to the model. The extracted features were examined using a heat map and principal components analysis (PCA) in MetaboAnalyst and R. Unlike conventional regression modelling, the penalized regression model employed in this paper does not assign a statistical significance (p) value for the extracted metabolite features. Instead glmnet includes bootstrapped cross-validation for tuning and selecting the optimal λ, as well as the selection of the penalization parameter (α), which penalizes the metabolite feature coefficients. Metabolite features whose penalized coefficients are > 0 are retained in the overall model in the elastic-net framework24.

## Conclusion

In this paper, we have demonstrated the use of elastic net regularized regression to study the degradation of the plasma metabolome under a sub-optimal freezing condition. Using this method, metabolites were ranked based on their temporal profile while simultaneously discarding metabolites whose temporal changes were negligible, allowing a more focused and objective selection of interesting metabolites. Furthermore, our data suggests that the plasma metabolome is relatively stable at −20 °C with few metabolites drastically changing until 1 month. However, a good compromise between storage temperature and plasma stability is seen at 12–15 days storage. This provides more information for logisticians to plan sampling and storage procedures for collecting biological samples from low-resource areas.

## Change history

• ### 15 June 2018

A correction to this article has been published and is linked from the HTML and PDF versions of this paper. The error has not been fixed in the paper.

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12. Determan, C. Optimal algorithm for metabolomics classification and feature selection varies by dataset. Int. J. Biol. 7 (2015).

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16. Bujak, R., Daghir-Wojtkowiak, E., Kaliszan, R. & Markuszewski, M. J. PLS-Based and Regularization-Based Methods for the Selection of Relevant Variables in Non-targeted Metabolomics Data. Frontiers in molecular biosciences 3, 35, https://doi.org/10.3389/fmolb.2016.00035 (2016).

17. Di Giovanni, V. et al. Metabolomic Changes in Serum of Children with Different Clinical Diagnoses of Malnutrition. J. Nutr.. https://doi.org/10.3945/jn.116.239145 (2016).

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24. Benton, M. C. et al. Methylome-wide association study of whole blood DNA in the Norfolk Island isolate identifies robust loci associated with age. Aging (Albany NY) 9, 753–766, https://doi.org/10.18632/aging.101187 (2017).

25. Gerstung, M. et al. Combining gene mutation with gene expression data improves outcome prediction in myelodysplastic syndromes. 6, 5901, https://doi.org/10.1038/ncomms6901 (2015).

26. Hinton, D. J. et al. Metabolomics biomarkers to predict acamprosate treatment response in alcohol-dependent subjects. Sci. Rep. 7, 2496, https://doi.org/10.1038/s41598-017-02442-4 (2017).

27. Rist, M. J. et al. Metabolite patterns predicting sex and age in participants of the Karlsruhe Metabolomics and Nutrition (KarMeN) study. PLOS ONE 12, e0183228, https://doi.org/10.1371/journal.pone.0183228 (2017).

28. Wishart, D. S. et al. HMDB 3.0–The Human Metabolome Database in 2013. Nucleic Acids Res. 41, D801–807, https://doi.org/10.1093/nar/gks1065 (2013).

29. Kalhan, S. C. et al. Plasma Metabolomic Profile in Non-Alcoholic Fatty Liver Disease. Metabolism 60, 404–413, https://doi.org/10.1016/j.metabol.2010.03.006 (2011).

30. Stanstrup, J., Schou, S. S., Holmer-Jensen, J., Hermansen, K. & Dragsted, L. O. Whey Protein Delays Gastric Emptying and Suppresses Plasma Fatty Acids and Their Metabolites Compared to Casein, Gluten, and Fish Protein. J. Proteome Res. 13, 2396–2408, https://doi.org/10.1021/pr401214w (2014).

31. Huang, X. et al. A New Strategy for Analyzing Time-Series Data Using Dynamic Networks: Identifying Prospective Biomarkers of Hepatocellular Carcinoma. 6, 32448, https://doi.org/10.1038/srep32448 (2016).

32. Diercks, D. B., Owen, K. P., Kline, J. A. & Sutter, M. E. Urine metabolomic analysis to detect metabolites associated with the development of contrast induced nephropathy. Clinical and experimental emergency medicine 3, 204–212, https://doi.org/10.15441/ceem.15.110 (2016).

33. Dai, D. et al. Time-resolved metabolomics analysis of individual differences during the early stage of lipopolysaccharide-treated rats. 6, 34136, https://doi.org/10.1038/srep34136 (2016).

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35. Friedman, J. H., Hastie, T. & Tibshirani, R. Regularization Paths for Generalized Linear Models via Coordinate Descent. 2010 33, 22, https://doi.org/10.18637/jss.v033.i01 (2010).

36. R Core Team. R: A language and environment for statistical computing. R Foundation for Statistical Computing, https://www.R-project.org/ (2017).

37. Kuhn, M. Building Predictive Models in R Using the caret Package. 2008 28, 26, https://doi.org/10.18637/jss.v028.i05 (2008).

## Acknowledgements

GB Gonzales is a postdoctoral fellow of the Research Foundation Flanders (FWO). The authors also acknowledge the Hercules infrastructure funding AUGE/13/13 for the Synapt G2Si HDMS. Special thanks also go to Celine Meerpoel, Karl De Ruyck and Mohamed Fathi Abdallah Abdelmohsen for the assistance in preparing the samples, and to Vladimer Kobayashi for technical assistance in building the models.

## Author information

Authors

### Contributions

G.B.G. conceived and designed the research project and performed the experiments and data analysis. G.B.G. and S.S. contributed to the writing of the manuscript.

### Corresponding author

Correspondence to Gerard Bryan Gonzales.

## Ethics declarations

### Competing Interests

The authors declare no competing interests.

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

## Rights and permissions

Reprints and Permissions

Gonzales, G.B., De Saeger, S. Elastic net regularized regression for time-series analysis of plasma metabolome stability under sub-optimal freezing condition. Sci Rep 8, 3659 (2018). https://doi.org/10.1038/s41598-018-21851-7

• Accepted:

• Published:

• DOI: https://doi.org/10.1038/s41598-018-21851-7

• ### From bedside to bench—practical considerations to avoid pre-analytical pitfalls and assess sample quality for high-resolution metabolomics and lipidomics analyses of body fluids

• Rainer Lehmann

Analytical and Bioanalytical Chemistry (2021)


text:
Prove a set is orthogonal.

1. Apr 3, 2012

roto25

How would you prove, using the integral product, that the set of {cos x, cos 2x, cos 3x, cos 4x, ...} is an orthogonal set?

Last edited: Apr 4, 2012
2. Apr 4, 2012

tiny-tim

welcome to pf!

hi roto25! welcome to pf!

i] define the integral product

ii] define orthogonal set

iii] apply i and ii …

what do you get?

3. Apr 5, 2012

Bavid

over the interval -pi to pi, the integral of cos(mx)cos(nx)dx is zero, as long as m and n are integers. Therefore, if you select ANY pair of elements from the set, the 'integral of their product' will be zero, thereby satisfying the condition of orthogonality.

4. Apr 5, 2012

chiro

On top of what Bavid said if you don't know where to start set up the integral and use integration by parts.


text:
January 1, 0001

# Reproducible Research Week 1

## Replication

The ultimate standard for strengthening scientific evidence is replication of finding and conducting studies with independent

• Investigators
• Data
• Analytical Methods
• Laboratories
• Instruments

Replication is particularly important in studies that can impact broad policy or regulatory decisions

### What’s wrong with replication?

Some studies cannot be replicated

• No time, opportunistic
• No money
• Unique

Reproducible Research: Make analytic data and code available so that others may reproduce findings

Reproducibility bridges the gap between replication which is awesome and doing nothing.

## Why do we need reproducible research?

New technologies increasing data collection throughput; data are more complex and extremely high dimensional

Existing databases can be merged into new “megadatabases”

Computing power is greatly increased, allowing more sophisticated analyses

For every field “X” there is a field “Computational X”

## Research Pipeline

Measured Data -> Analytic Data -> Computational Results -> Figures/Tables/Numeric Summaries -> Articles -> Text

The computer code and fully specified computational procedures used for development of the candidate omics-based test should be made sustainably available

“Ideally, the computer code that is released will encompass all of the steps of computational analysis, including all data preprocessing steps. All aspects of the analysis needs to be transparently reported” – IOM Report

### What do we need for reproducible research?

• Analytic data are available
• Analytic code are available
• Documentation of code and data
• Standard means of distribution

### Who is the audience for reproducible research?

Authors:

• Want to make their research reproducible
• Want tools for reproducible research to make their lives easier (or at least not much harder)

• Want to reproduce (and perhaps expand upon) interesting findings
• Want tools for reproducible research to make their lives easier.

### Challenges for reproducible research

• Authors must undertake considerable effort to put data/results on the web (may not have resources like a web server)
• Readers must download data/results individually and piece together which data go with which code sections, etc.
• Readers may not have the same resources as authors
• Few tools to help authors/readers

### What happens in reality

Authors:

• Just put stuff on the web
• (Infamous for disorganization) Journal supplementary materials
• There are some central databases for various fields (e.g biology, ICPSR)

• Piece together the software and run it

## Literate (Statistical) Programming

An article is a stream of text and code

Analysis code is divided into text and code “chunks”

Each code chunk loads data and computes results

Presentation code formats results (tables, figures, etc.)

Article text explains what is going on

Literate programs can be weaved to produce human-readable documents and tagled to produce machine-readable documents

Literate programming is a general concept that requires

1. A documentation language (human readable)
2. A programming language (machine readable)

Knitr is an R package that brings a variety of documentation languages such as Latex, Markdown, and HTML

### Quick summary so far

Reproducible research is important as a minimum standard, particularly for studies that are difficult to replicate

Infrastructure is needed for creating and distributing reproducible document, beyond what is currently available

There is a growing number of tools for creating reproducible documents

Golden Rule of Reproducibility: Script Everything

## Steps in a Data Analysis

1. Define the question
2. Define the ideal data set
3. Determine what data you can access
4. Obtain the data
5. Clean the data
6. Exploratory data analysis
7. Statistical prediction/modeling
8. Interpret results
9. Challenge results
10. Synthesize/write up results
11. Create reproducible code

“Ask yourselves, what problem have you solved, ever, that was worth solving, where you knew all of the given information in advance? Where you didn’t have a surplus of information and have to filter it out, or you had insufficient information and have to go find some?” – Dan Myer

Defining a question is the kind of most powerful dimension reduction tool you can ever employ.

### An Example for #1

Can I automatically detect emails that are SPAM or not?

Make it concrete

Can I use quantitative characteristics of emails to classify them as SPAM?

### Define the ideal data set

The data set may depend on your goal

• Descriptive goal – a whole population
• Exploratory goal – a random sample with many variables measured
• Inferential goal – The right population, randomly sampled
• Predictive goal – a training and test data set from the same population
• Causal goal – data from a randomized study
• Mechanistic goal – data about all components of the system

### Determine what data you can access

Sometimes you can find data free on the web

Other times you may need to buy the data

If the data don’t exist, you may need to generate it yourself.

### Obtain the data

Try to obtain the raw data

Be sure to reference the source

Polite emails go a long way

If you load the data from an Internet source, record the URL and time accessed

### Clean the data

Raw data often needs to be processed

If it is pre-processed, make sure you understand how

Understand the source of the data (census, sample, convenience sample, etc)

May need reformatting, subsampling – record these steps

Determine if the data are good enough – If not, quit or change data

### Exploratory Data Analysis

Look at summaries of the data

Check for missing data

-> Why is there missing data?

Look for outliers

Create exploratory plots

Perform exploratory analyses such as clustering

If it’s hard to see your plots since it’s all bunched up, consider taking the log base 10 of an axis

plot(log10(trainSpan$capitalAve + 1) ~ trainSpam$type)

### Statistical prediction/modeling

Should be informed by the results of your exploratory analysis

Exact methods depend on the question of interest

Transformations/processing should be accounted for when necessary

Measures of uncertainty should be reported.

### Interpret Results

Use the appropriate language

• Describes
• Correlates with/associated with
• Predicts

Gives an explanation

Interpret Coefficients

Interpret measures of uncertainty

### Challenge Results

Challenge all steps:

• Question
• Data Source
• Processing
• Analysis
• Conclusions

Challenge measures of uncertainty

Challenge choices of terms to include in models

Think of potential alternative analyses

### Synthesize/Write-up Results

Summarize the analyses into the story

Don’t include every analysis, include it

• If it is needed for the story
• If it is needed to address a challenge
• Order analyses according to the story, rather than chronologically
• Include “pretty” figures that contribute to the story

### In the lecture example…

 Can I use quantitative characteristics of the emails to classify them as SPAM?

Describe the approach

 Collected data from UCI -> created training/test sets

 Explored Relationships

 Choose logistic model on training set by cross validation

 Applied to test, 78% test set accuracy

Interpret results

 Number of dollar signs seem reasonable, e.g. “Make more money with Viagra  ”

Challenge Results

 78% isn’t that great

 Could use more variables

 Why use logistic regression?

## Data Analysis Files

Data

• Raw Data
• Processed Data

Figures

• Exploratory Figures
• Final Figures

R Code

• Raw/Unused Scripts
• Final Scripts
• R Markdown Files

Text

• Text of Analysis/Report

### Raw Data

Should be stored in the analysis folder

If accessed from the web, include URL, description, and date accessed in README

### Processed Data

Processed data should be named so it is easy to see which script generated the data

The processing script – processed data mapping should occur in the README

Processed data should be tidy

### Exploratory Figures

They do not need to be “pretty”

### Final Figures

Usually a small subset of the original figures

Axes/Colors set to make the figure clear

Possibly multiple panels

### Raw Scripts

May be multiple versions

May include analyses that are later discarded

### Final Scripts

Clearly commented

• Small comments liberally - what, when, why, how

• Bigger commented blocks for whole sections

Include processing details

Only analyses that appear in the final write-up

### R Markdown Files

R Markdown files can be used to generate reproducible reports

Text and R code are integrated

Very easy to create in RStudio

Not necessary if you use R Markdown

Should contain step-by-step instructions for analysis

### Text of the document

It should contain a title, introduction (motivation), methods (statistics you used), results (including measures of uncertainty), and conclusions (including potential problems)

It should tell a story

It should not include every analysis you performed

References should be included for statistical methods


text:
# Is magnetic field due to an electric current a relativistic effect?

I was reading a paper of the same name by Oleg D. Jefimenko; here is the concerned text:

[...] relativistic force transformation equations demand the presence of an electric field when the interactions between electric charges are assumed to be entirely due to a magnetic force. We could interpret this result as the evidence that the electric field is a relativistic effect. But the well known fact that similar calculations demand the presence of a magnetic field, if the interactions between the charges are assumed to be entirely due to an electric force, makes such an interpretation impossible (unless we are willing to classify both the magnetic and the electric field as relativistic effects, which is absurd).

Both fields—the electric field and the magnetic field are necessary to make interactions between electric charges relativistically correct.By inference then, any force field compatible with the relativity theory must have an electric-like ‘subfield’ and a magnetic-like ‘subfield’.

I, being a total novice, have no expertise in special relativity. But I read that magnetism is really due to Coulomb's Law.

This is taken from Chris White's answer here:

[...] magnetism is nothing more than electrostatics combined with special relativity.

And this is taken from the horse's mouth; from Purcell's Electricity and Magnetism:

[...] the magnetic interaction of electric currents can be recognized as an inevitable corollary to Coulomb's Law. If the postulates of relativity are valid, if electric charge is invariant, and if Coulomb's law holds, then the effects we commonly call "magnetic" are bound to occur.

These two statements speak volumes to the fact that magnetic field is due to electrostatics and relativity; which ought make it a relativistic effect, isn't it?

then why did Jefimenko tell otherwise?

He did deduce in that paper the electric field is due to the magnetic interaction combined with relativity!

Prior to the reading of that paper, I bore in mind that if magnetic interaction occurs due to Coulomb's Law, then electric field is more fundamental.

But, this is completely contrary to what Jefimenko said; if I consider the interaction between charges to be magnetic, then the relativistic force transformation equation makes sure the existence of electric fields; so magnetic field is more fundamental?

My questions are:

$\bullet$ Is Jefimenko contradicting the statement of Purcell? What did he actually want to say?

$\bullet$ What should be the actual interpretation if it is wrong to assert that 'magnetism is nothing more than electrostatics combined with special relativity'?

• This has been asked and answered many times, see this for example, and the several "related" questions to the right. Short answer: one has to read what's written on this topic very carefully, as misinterpretations have occurred. Read the comments to White's answer. I don't see anything wrong with Jefimenko's statement, but there might be some subtly that I'm missing. – garyp Jan 18 '16 at 3:24
• @garyp: Yes, I've checked this before posting the query but really can you tell me what is actually happening? Who is right? Actually, I'm seeing no one is wrong; I believed White and Purcell till Jefimenko proved the electric field from the fact that the real interaction is magnetic. Lastly, he said, both are not relativistic effects as this would be absurd; why? What should be the actual interpretation? Why can't simply Jefimenko accept Purcell? – user36790 Jan 18 '16 at 3:29
• – Art Brown Jan 19 '16 at 6:44

I don't think Jefimenko is unclear in the slightest. You have some choices:

1. given magnetic forces first, add SR and deduce electric forces, or
2. given electric forces first, add SR and deduce magnetic forces, or
3. learn SR first and expect to have electromagnetic forces from the get go then add the Lorentz Force.

I think it's wishful thinking to imagine you could describe anything at all as blah+SR and gain understanding while also not learning SR.

So first lets start by understanding relativistic kinematics and dynamics and then lets look at Maxwell from the viewpoint where we understand SR.

OK. So you have a 4d space-time, made up of events. You have curves which are functions from intervals of $\mathbb R$ into space-time. And there might be many parametrizations.

For instance the function from $\mathbb R$ into $\mathbb R\times \mathbb R^3$ that sends $t$ to $(tc,\vec 0)$ is one parametrization. And the function from $\mathbb R$ into $\mathbb R\times \mathbb R^3$ that sends $t$ to $(5tc,\vec 0)$ is another parametrization of the same path. It still goes along the $t$ axis but now a unit distance in $\mathbb R$ does not correspond to a unit proper time along the path.

Also not that the curve has a tangent everywhere along it and that the tangent is time-like (and future pointing). In general, when a curve corresponds to the motion of an object you can consider the instantaneously co-moving inertial frame to be the global inertial frame in SR that corresponds to the tangent of the curve at that event.

In that frame the object is instantaneously at rest. And you can consider parametrizing the curve so the unit tangent points in the direction of the energy momentum four vector, and thus we can scale the unit tangent by the (rest) mass of the object and get the energy momentum four vector.

So now if we have the curve and the mass, we effectively have the energy-momentum too.

So really, now we can look at what forces do. The unit tangent parametrized by proper time now determines the kinematics. Because we effectively know which direction in space-time it is going by knowing the tangent. The unit tangent gives the direction in space-time.

And so if the object always has an instantaneously co-moving inertial frame, then that tangent is always pointing towards the future and is a time-like vector. And if it is the unit tangent (or the rest mass times that) then really it is a future pointing unit tangent vector turning into another future pointing unit tangent vector.

So really all forces could possibly do is rotate a future pointing unit tangent vector into another future pointing unit tangent vector, at a certain rate as given either by an inertial frame, or at a certain rate per proper time along the curve.

So now describing changing kinematics, with rate of change means giving a rotation as per time rate that tangent vector rotates. And there are different independent rotations in 4d, three boosts and three rotations. Everyone agrees on the need for six generators, though what one frame considers a pure boost would not be a pure boost for a different frame.

So we now expect all relativistic forces to be described by a rank two tensor. And that is just how the force describes a rotation of the unit tangent.

The unit tangent is the kinematics, and the rotations reflect that a unit tangent turning into a slightly different unit tangent is a rotation of the first tangent.

Now we can get to Jefimenko. You could start with saying how your magnetic forces act in a frame, so you would describe some rotations for tangents that don't start in one fixed direction and then find out how those work.

Or you could start with tangents that start in that direction (electric forces) and figure out the others.

Why you'd want to start with just one and get the other is unknown. Historical reasons could be justified but otherwise it would be better to sit down and learn SR and note that forces should rotate unit tangents.

• Hmmm... thanks for answering... however, I couldn't make anything out of it (this is the first answer of yours that I couldn't understand:( ). Could you tell me what you meant by parameterizing the curve; you've used this term frequently; could you please make it clear? – user36790 Jan 19 '16 at 5:13


text:
# What is the color of Cu^(2+) in solution? What is the color of glucose in solution? What the color and formula are of copper (l) oxide?

Jan 26, 2017

The $\text{cupric ion}$ has a truly beautiful blue colour in aqueous solution.....
${\left[C u {\left(O {H}_{2}\right)}_{6}\right]}^{2 +}$ gives a blue coloured solution in water:
Very large, macroscopic cyrstals of $\left[C u {\left(O {H}_{2}\right)}_{6}\right] S {O}_{4}$ can be grown from these solutions:
And when glucose is dissolved in water, a colourless solution results. On the other hand, $\text{cuprous oxide}$, i.e. $C {u}_{2} O$, is a brick-red powder, that is insoluble in water.


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# Boost your Octave (for MATLAB users)

Let’s face it, you are using Octave because MATLAB is too expensive for you. Maybe you’re not student anymore and got used to play with MATLAB, but now you need the best free alternative.

Certainly, if money is not a problem, MATLAB would overperform Octave in most of their characteristics (not all of them and I’ll show you) and is surely the best option.

Fear not! GNU Octave is here, and yes, it is indeed the best alternative to MATLAB, but… it’s not the same. We know, and here I present some useful tips to avoid any conflict with your migration to the Octave world.

# Opera: Tijuana style

Ciudad Juárez, Matamoros, Nuevo Laredo, Reynosa… what do these cities have in common? They are border cities, where the violence in Mexico exploded due to the War on Drugs that devastated the northern part of the country.

Tijuana is precisely one of those cities that felt it heavily, but today teaches us a very nice lesson. Since years (even before the War on Drugs) Tijuana was world-known for its careless way of life. Alcohol, drugs, women, nasty streets and tough night life were part of the imagination when someone imagined the city.

The inhabitants, however, never lost their faith and found new ways to fight that feeling. As true Mexicans, they knew that violence cannot be stopped with more violence, and thus fought it the way they knew: sing and smile.

The Tijuana Cultural Centre and their young talents decided to give hope and draw smiles in its inhabitants with their own version of street Opera, a movement that has been seen previously in Europe, but a movement that Tijuana needed to gain energy and encourage their people to not give up and live peacefully. Did it work? Too soon to say, but oh boy! truly inspires.

# Día de Muertos: Tips to enjoy the tradition

Probably you know a bit about this tradition due to the curiosity about the “celebration of Death”. Well, Death is a topic with different points of view in the world.

As a Mexican there are many things that still surprise about this festivity, mainly because it is celebrated in different ways all along Mexico. Oh yeah, last year I made a video with the historical background and main topics of it. I recommend you to watch it before proceeding with the lines below.

I assume that you now understood that Mexicans are not celebrating Death, in the same sense that we are not having parties to say good-bye to our beloved dead. Continue reading

# The unforgettable October the 2nd

October the 2nd is a day making history in Mexico. Mexicans look at it with disgust and sadness, adding one painful memory to the national feeling.

1968 was the year of worldwide awakening and cultural revolutions led by students and social activists, and Mexico was not the exception. It was also the year in which the Olympics were held in Mexico, where the eyes of the world were pointing to the country. Continue reading

# How to use polyval

One of the easiest and most useful tools in Matlab is polyval, a very nice function that evaluates a polynomial function given its parameters and the range to evaluate… huh? All right, all right, we wanna be clear here, right?

Suppose we are given a polynomial function, let’s say:

$f(x) = 3x^3 + 4x + 2$

and we would like to represent it in Matlab like that. Well, we cannot just write it like that. Remember Matlab is a numerical computation programm, which means, that it won’t compute any symbol. So forget it if you wanna computate something writting letters. Matlabs wants only numbers. You might put names to the variables, but still, Matlab computes only with numbers.

Now, what to do? There’s where our great friend polyval comes to the rescue! Continue reading

# Cinco de Mayo: The battle

Without being a stereotype per se, the famous “May 5th” is loud and internationally linked to a Mexican celebration, but… what is supposed to be celebrated?

I’ll be short: the battle where Mexico defeated France. That’s all. In the second french intervention in Mexico (yes, they were two major french interventions in Mexico) in the middle of the XIX century, France invaded Mexico to annex it as a satellite state of France. The invasion started in Veracruz and headed to Mexico City, but in the middle of the way (in Puebla) the french army was slammed on the brakes and defeated by a totally underrated Mexican army.

# Treasure Hunt in Google Maps

And the guys in Google seem to have lots of free time. Following the concept they created last year with an 8-bit Google Maps, this year they come with a new idea for April’s fool: an old style Treasure Hunt. I don’t know for how long it’s gonna be available but this trick looks very odd (tho funny).

When you open Google Maps for the first time you can see on the upper right options of visualization, an option that states ” Treasure”. Click it. That’s it, you can see now your map as an old middle aged map. They say you can find some hidden treasure chests, but I think it’d be very hard and time-consuming.

But what happens when you wanna have a closer look? Well, unfortunately it zooms in until a definite distance, as they didn’t rendered all buildings and streets, just the main ones. However, if you choose the “Street view”, you will have a surprise having an old telescope-like look.

Aaaaand, that’s this year’s Easter Egg from Google Maps. Maybe not so special but, quite creative and entertaining. Have a look at your city. How much stuff is in it? Did you see the most interesting things?

# Naquito Abroad: The Oktoberfest Challenge

Finally my first video of “Naquito Abroad” is out. And after days of editing, I finally got some material out. The language of that series of videos is most of the time Spanish. But, as in my other videos, you can find the subtitles in English. Actually, a big portion of the video is also in English (when I was rambling in the tents of Oktoberfest).

This is the product of a day in the biggest Beer festival in the world. I show you here the history and organization of the Oktoberfest, so you can have a look into the event as everyone else.

For a deeper and more detailed information of it, you can read it (although in Spanish) in my other post “Bienvenido al Oktoberfest“.

So, come this year, but in the meantime, Proooooost!


text:
# maven-users mailing list archives

##### Site index · List index
Message view
Top
From "Aleksandr Shneyderman" <a...@law.columbia.edu>
Subject RE: Building Webapps
Date Thu, 12 Aug 2004 20:02:37 GMT
```
Write appropriate filter not to include tons of stuff

**/*

is just one example which obviously is not what you need.

-----Original Message-----
From: David Erickson [mailto:derickson@cmcflex.com]
Sent: Thursday, August 12, 2004 3:47 PM
To: Maven Users List; alex@law.columbia.edu
Subject: Re: Building Webapps

But then that will include tons of stuff I dont want that is within my
src directory, ie i dont want my .java files going along with it.. nor
my test resources etc etc.. -David
----- Original Message -----
From: "Alex Shneyderman" <alex@law.columbia.edu>
To: "'Maven Users List'" <users@maven.apache.org>;
<alex@law.columbia.edu>
Sent: Thursday, August 12, 2004 1:44 PM
Subject: RE: Building Webapps

> Sorry, of course no targetPath element
>
> > -----Original Message-----
> > From: Alex Shneyderman [mailto:alex@law.columbia.edu]
> > Sent: Thursday, August 12, 2004 3:43 PM
> > To: 'Maven Users List'
> > Subject: RE: Building Webapps
> >
> >
> > You can change it to
> >
> >       <resource>
> >         <directory>\${basedir}/src</directory>
> >         <targetPath>resources</targetPath>
> >         <includes>
> >           <include>**/*</include>
> >         </includes>
> >       </resource>
> >
> > to achieve that.
> >
> > Alex.
> >
> > > -----Original Message-----
> > > From: David Erickson [mailto:derickson@cmcflex.com]
> > > Sent: Thursday, August 12, 2004 3:39 PM
> > > To: Maven Users List; epugh@upstate.com
> > > Subject: Re: Building Webapps
> > >
> > > Eric is it possible to specify the target location for the
> > > resource
> to
> > be
> > > included at?  I guess thats kinda vague, but an example is:
> > >       <resource>
> > >         <directory>\${basedir}/src/resources</directory>
> > >         <targetPath>resources</targetPath>
> > >         <includes>
> > >           <include>*</include>
> > >         </includes>
> > >       </resource>
> > >
> > > so everything within src/resources gets put into /resources on the

> > > classpath -David
> > >
> > > ----- Original Message -----
> > > From: "Eric Pugh" <epugh@upstate.com>
> > > To: "Maven Users List" <users@maven.apache.org>
> > > Sent: Thursday, August 12, 2004 1:25 PM
> > > Subject: RE: Building Webapps
> > >
> > >
> > > > The solution is simple!
> > > >
> > > > From
> > http://maven.apache.org/reference/plugins/eclipse/properties.html:
> > > > maven.eclipse.classpath.include=src\resources
> > > >
> > > > Eric
> > > >
> > > > > -----Original Message-----
> > > > > From: David Erickson [mailto:derickson@cmcflex.com]
> > > > > Sent: Thursday, August 12, 2004 9:16 PM
> > > > > To: Maven User List
> > > > > Subject: Building Webapps
> > > > >
> > > > >
> > > > > Hey everyone I have been having a difficult time figuring out
> the
> > best
> > > > > process to work with maven when building webapps.  Here is how

> > > > > I
> > > started:
> > > > > Eclipse 3, with myEclipseIDE and of course maven.  I had my
> webapp
> > > project
> > > > > and another non-webapp project.  This was working pretty well,

> > > > > I had eclipse set the nonwebapp project as dependent to my
> > > > > webapp, so whenever i deployed
> > > > > it would jar up the classes and chuck them into my web-inf/lib
> > folder
> > > and
> > > > > everything was great.  All up until I found that I needed a
> > > > > resource from my webapp to be placed on the class path, so
> > > > > something like \src\resources\myContext.xml (Spring) needed to

> > > > > be sitting in \WEB-INF\classes, or the root of my classes jar
> > > > > file.  Well
> maven
> > > handles
> > > > > this easily by including it in the jar resources section.
> However
> > > Eclipse
> > > > > the only way to do this (as far as I am aware) is to put it
> > > > > into
> > my
> > > > > \src\java directory and have it copy it in with my classes.
> > > > > And
> I
> > DO
> > > NOT
> > > > > like doing that because basically its just a hack.  Ideally
> > > > > I'd like to have maven rebuild stuff anytime there are
> > > > > changes, kind of like
> > eclipse's
> > > auto
> > > > > build, however I don't know if thats feasible?  It just seems
> like
> > > your
> > > > > fighting two battles using maven and an IDE, the first battle
> > > > > is
> > to
> > > get
> > > > > maven to work correctly in the commandline, and the second
> battle
> > is
> > > to
> > > > > attempt to have your ide do what maven does for includes and
> > > > > stuff.  I know you can execute maven targets from within
> > > > > eclipse, and I've
> tried
> > > > > that, but
> > > > > it gets annoying having to constantly click the targets every
> time
> > you
> > > > > change anything so you can see how it looks in your
> > > > > webbrowser.
> > > > >
> > > > > Anyway I guess that was a lot of random thoughts put
> > > > > together..
> I
> > > > > guess I'm
> > > > > just curious how everyone else is doing this, I'm open to
> > switching to
> > > > > IntelliJ if i has a lot better support for this kind of
> > > > > thing.. eclipse has been driving me nuts lately.
> > > > > Thanks all,
> > > > > David
> > > > >
> > > > >
> > > > >
> > --------------------------------------------------------------------
> > -
> > > > > To unsubscribe, e-mail: users-unsubscribe@maven.apache.org
> > > > > For additional commands, e-mail: users-help@maven.apache.org
> > > >
> > > >
> > > >
> > --------------------------------------------------------------------
> > -
> > > > To unsubscribe, e-mail: users-unsubscribe@maven.apache.org
> > > > For additional commands, e-mail: users-help@maven.apache.org
> > > >
> > > >
> > >
> > >
> > >
> ---------------------------------------------------------------------
> > > To unsubscribe, e-mail: users-unsubscribe@maven.apache.org
> > > For additional commands, e-mail: users-help@maven.apache.org
> >
> >
> >
> > --------------------------------------------------------------------
> > -
> > To unsubscribe, e-mail: users-unsubscribe@maven.apache.org
> > For additional commands, e-mail: users-help@maven.apache.org
>
>
>
> ---------------------------------------------------------------------
> To unsubscribe, e-mail: users-unsubscribe@maven.apache.org
> For additional commands, e-mail: users-help@maven.apache.org
>
>

---------------------------------------------------------------------
To unsubscribe, e-mail: users-unsubscribe@maven.apache.org


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### Archive

Posts Tagged ‘miklos schweitzer’

## Problems of the Miklos Schweitzer Competition 2014

November 10, 2014 5 comments

Problem 1. Let ${n}$ be a positive integer. Let ${\mathcal{F}}$ be a familiy of sets that contains more than half of all subsets of an ${n}$-element set ${X}$. Prove that from ${\mathcal{F}}$ we can select ${\lceil \log_2 n\rceil+1 }$ sets that form a separating family of ${X}$, i.e., for any two distinct elements of ${X}$ there is a selected set containing exactly one of the two elements.

Problem 2. let ${k \geq 1}$ and let ${I_1,...,I_k}$ be non-degenerate subintervals of the interval ${[0,1]}$. Prove that

$\displaystyle \sum \frac{1}{|I_i \cup I_j|} \geq k^2,$

where the summation is over all pairs of indices ${(i,j)}$ such that ${I_i}$ and ${I_j}$ are not disjoint.

Problem 3. We have ${4n+5}$ points in the plane, no three of them collinear. The points are colored with two colors. Prove that from the points we can form ${n}$ empty triangles (they have no colored points in their interiors) with pairwise disjoint interiors, such that all points occuring as vertices of the ${n}$ triangles have the same color.

Problem 4. For a positive integer ${n}$, let ${f(n)}$ be the number of sequences ${a_1,...,a_k}$ of positive integers such that ${a_i \geq 2}$ and ${a_1...a_k = n}$ for ${k \geq 1}$. We make the convention ${f(1)=1}$. Let ${\alpha}$ be the unique real number greater than ${1}$ such that ${\sum_{n=1}^\infty n^{-\alpha}=2}$. Prove that

• (i) ${ \sum_{ k = 1}^n f(k)= O(n^\alpha)}$.
• (ii) There exists no number ${\beta<\alpha}$ such that ${f(n)=O(n^\beta)}$.

Problem 5. Let ${\alpha}$ be a non-real algebraic integer of degree two, and let ${P}$ be the set of irreducible elements of the ring ${\Bbb{Z}[\alpha]}$. Prove that

$\displaystyle \sum_{ p \in P} \frac{1}{|p|^2} = \infty.$

Problem 6. Let ${\rho : G \rightarrow GL(V)}$ be a representation of a finite ${p}$-group ${G}$ over a field of characteristic ${p}$. Prove that if the restriction of the linear map ${\sum_{ g \in G} \rho(g)}$ to a finite dimensional subspace ${W}$ of ${V}$ is injective, then the subspace spanned by the subspaces ${\rho(g)W}$ (${g \in G}$) is the direct sum of these subspaces.

Problem 7. Lef ${f: \Bbb{R} \rightarrow \Bbb{R}}$ be a continuous function and let ${g: \Bbb{R} \rightarrow \Bbb{R}}$ be arbitrary. Suppose that the Minkowski sum of the graph of ${f}$ and the graph of ${g}$ (i.e. the set ${\{(x+y,f(x)+g(y) : x,y \in \Bbb{R}\}}$ has Lebesgue measure zero. Does it follow then that the function ${f}$ is of the form ${f(x)=ax+b}$, with suitable constants ${a,b \in \Bbb{R}}$?

Problem 8. Let ${n \geq 1}$ be a fixed integer. Calculate the distance

$\displaystyle \inf_{p,f} \max_{0 \leq x \leq 1} |f(x)-p(x)|,$

where ${p}$ runs over polynomials of degree less than ${n}$ with real coefficients and ${f}$ runs over functions of the form

$\displaystyle f(x) = \sum_{ k = n}^\infty c_kx^k$

defined on the closed interval ${[0,1]}$, where ${c_k\geq 0}$ and ${\sum_{k=n}^\infty c_k =1}$.

Problem 9. Let ${\rho : \Bbb{R}^n \rightarrow \Bbb{R},\ \rho(x)=e^{-\|x\|^2}}$, and let ${K \subset \Bbb{R}^n}$ be a convex body, i.e. a compact convex set with nonempty interior. Define the barycenter ${s_K}$ of the body ${K}$ with respect to the weight function ${\rho}$ by the usual formula

$\displaystyle s_K = \frac{\int_K \rho(x) x dx}{\int_K \rho(x)dx}.$

Prove that the translates of the body ${K}$ have pairwise distinct barycenters with respect to ${\rho}$.

Problem 10. To each vertex of a given triangulation of the two dimensional sphere, we assign a convex subset of the plane. Assume that the three convex sets corresponding to the three vertices of any two dimensional face of the triangulation have at least one point in common. Show that there exist four vertices such that the corresponding convex sets have at least one point in common.

Problem 11. Let ${U}$ be a random variable that is uniformly distributed on the interval ${[0,1]}$, and let

$\displaystyle S_n = 2\sum_{k=1}^n \sin(2kU\pi).$

Show that, as ${n \rightarrow \infty}$, the limit distribution of ${S_n}$ is the Cauchy distribution with density function ${f(x) =\frac{1}{\pi(1+x^2)}}$.

## Miklos Schweitzer 2013 Problem 7

November 22, 2013 1 comment

Problem 7. Suppose that ${f: \Bbb{R} \rightarrow \Bbb{R}}$ is an additive function (that is ${f(x+y) = f(x)+f(y)}$ for all ${x, y \in \Bbb{R}}$) for which ${x \mapsto f(x)f(\sqrt{1-x^2})}$ is bounded of some nonempty subinterval of ${(0,1)}$. Prove that ${f}$ is continuous.

Miklos Schweitzer 2013

Categories: Algebra, Analysis, Geometry

## Miklos Schweitzer 2013

November 9, 2013 2 comments

Problem 1. Let ${q \geq 1}$ be an integer. Prove that there exists an integer ${C_q}$ such that for any finite set ${A}$ containing only integers we have

$\displaystyle |A+qA| \geq (q+1)|A|-C_q.$

(${A+qA}$ is the set of integers of the form ${a+qa'}$ where ${a,a' \in A}$)

Problem 2. Prove that there is a constant ${k_0}$ such that the diophantine equation

$\displaystyle a^{2n}+b^{4n}+2013=ka^n b^{2n}$

has no positive integer solutions ${(a,b,n)}$ for ${k \geq k_0}$. (here I guess $k$ needs to be an integer, or else the conclusion is clearly false)

Problem 3. What are the numbers ${n}$ for which the ${A_n}$ alternating group has a permutation which is contained in exactly one ${2}$-Sylow subgroup of ${A_n}$?

Problem 4. Let ${A}$ be an Abelian group with ${n}$ elements. Prove that there are two subgroups in ${GL(n,\Bbb{C})}$, isomorphic to ${S_n}$, whose intersection is isomorphic to the automorphism group of ${A}$.

Problem 5. A subalgebra ${\mathfrak h}$ of a Lie algebra ${\mathfrak g}$ is said to have tha ${\gamma}$ property with respect to a scalar product ${\langle .,.\rangle}$ given on ${\mathfrak g}$ if ${X \in \mathfrak{h}}$ implies ${\langle [X,Y],X\rangle =0}$ for all ${Y \in \mathfrak g}$. Prove that the maximum dimension of ${\gamma}$-property subalgebras of a given ${2}$ step nilpotent Lie algebra with respect to a scalar product is independent of the selection of the scalar product.

Problem 6. Let ${\mathcal A}$ be a ${C^*}$ algebra with a unit element and let ${\mathcal A_+}$ be the cone of the positive elements of ${\mathcal A}$ (this is the set of such self adjoint elements in ${\mathcal A}$ whose spectrum is in ${[0,\infty)}$. Consider the operation

$\displaystyle x \circ y =\sqrt{x}y\sqrt{x},\ x,y \in \mathcal A_+.$

Prove that if for all ${x,y \in \mathcal A_+}$ we have

$\displaystyle (x\circ y)\circ y = x \circ (y \circ y),$

then ${\mathcal A}$ is commutative.

Problem 7. Suppose that ${f: \Bbb{R} \rightarrow \Bbb{R}}$ is an additive function (that is ${f(x+y) = f(x)+f(y)}$ for all ${x, y \in \Bbb{R}}$) for which ${x \mapsto f(x)f(\sqrt{1-x^2})}$ is bounded of some nonempty subinterval of ${(0,1)}$. Prove that ${f}$ is continuous. [Solution]

Problem 8. Let ${f : \Bbb{R} \rightarrow \Bbb{R}}$ be a continuous and strictly increasing function for which

$\displaystyle f^{-1}\left(\frac{f(x)+f(y)}{2}\right)(f(x)+f(y)) =(x+y)f\left(\frac{x+y}{2}\right)$

for all ${x,y \in \Bbb{R}}$ (${f^{-1}}$ denotes the inverse of ${f}$). Prove that there exist real constants ${a \neq 0}$ and ${b}$ such that ${f(x)=ax+b}$ for all ${x \in \Bbb{R}}$. [Solution]

Problem 9. Prove that there is a function ${f: (0,\infty) \rightarrow (0,\infty)}$ which is nowhere continuous and for all ${x,y \in (0,\infty)}$ and any rational ${\alpha}$ we have

$\displaystyle f\left( \left(\frac{x^\alpha+y^\alpha}{2}\right)^{\frac{1}{\alpha}}\right)\leq \left(\frac{f(x)^\alpha +f(y)^\alpha }{2}\right)^{\frac{1}{\alpha}}.$

Is there such a function if instead the above relation holds for every ${x,y \in (0,\infty)}$ and for every irrational ${\alpha}$.

Problem 10. Consider a Riemannian metric on the vector space ${\Bbb{R}^n}$ which satisfies the property that for each two points ${a,b}$ there is a single distance minimising geodesic segment ${g(a,b)}$. Suppose that for all ${a \in \Bbb{R}^n}$, the Riemannian distance with respect to ${a}$, ${\rho_a : \Bbb{R}^n \rightarrow \Bbb{R}}$ is convex and differentiable outside of ${a}$. Prove that if for a point ${x \neq a,b}$ we have

$\displaystyle \partial_i \rho_a(x)=-\partial_i \rho_b(x),\ i=1..n$

then ${x}$ is a point on ${g(a,b)}$ and conversely.

Problem 11. (a) Consider an ellipse in the plane. Prove that there exists a Riemannian metric which is defined on the whole plane, and with respect to which the ellipse is a geodesic. Prove that the Gaussian curvature of any such Riemannian metric takes a positive value.

(b) Consider two nonintersecting, simple closed smooth curves in the plane. Prove that if there is a Riemmanian metric defined on the whole plane and the two curves are geodesics of that metric, then the Gaussian curvature of the metric vanishes somewhere.

Problem 12. There are ${n}$ tokens in a pack. Some of them (at least one, but not all) are white and the rest are black. All tokens are extracted randomly from the pack, one by one, without putting them back. Let ${X_i}$ be the ratio of white tokens in the pack before the ${i}$-th extraction and let

$\displaystyle T =\max \{ |X_i-X_j| : 1 \leq i \leq j \leq n\}.$

Prove that ${\Bbb{E}(T) \leq H(\Bbb{E}(X_1))}$, where ${H(x)=-x\ln x -(1-x)\ln(1-x)}$.

(Thanks again to Eles Andras for the translation of the problems from Hungarian to English)

## Commutators and the identity matrix – Miklos Schweitzer 2012 Problem 6

January 8, 2013 1 comment

Suppose ${A,B,C}$ are complex ${n\times n}$ matrices such that ${[A,B]=C,\ [B,C]=A}$ and ${[C,A]=B}$, where ${[X,Y]=XY-YX}$ is the usual commutator. Prove that ${e^{4\pi A}}$ is the identity matrix.

Miklos Schweitzer 2012 Problem 6

## Miklos Schweitzer 2012 Problems

January 4, 2013 3 comments

1. Is there any real number ${\alpha}$ for which there exist two functions ${f,g: \Bbb{N} \rightarrow \Bbb{N}}$ such that

$\displaystyle \alpha=\lim_{n \rightarrow \infty} \frac{f(n)}{g(n)},$

but the function which associates to ${n}$ the ${n}$-th decimal digit of ${\alpha}$ is not recursive?

2. Call a subset ${A}$ of the cyclic group ${(\Bbb{Z}_n,+)}$ rich if for all ${x,y \in \Bbb{Z}_n}$ there exists ${r \in \Bbb{Z}_n}$ such that ${x-r,x+r,y-r,y+r}$ are all in ${A}$. For what ${\alpha}$ is there a constant ${C_\alpha>0}$ such that for each odd positive integer ${n}$, every rich subset ${A \subset \Bbb{Z}_n}$ has at least ${C_\alpha n^\alpha}$ elements?

3. Prove that if a ${k}$-chromatic graph’s edges are coloured in two colors in any way, there is a subtree with ${k}$ vertices and edges of the same color.

4. Let ${K}$ be a convex shape in the ${n}$ dimensional space, having unit volume. Let ${S \subset K}$ be a Lebesgue measurable set with measure at least ${1-\varepsilon}$, where ${0<\varepsilon<1/3}$. Prove that dilating ${K}$ from its centroid by the ratio of ${2\varepsilon \ln(1/\varepsilon)}$, the shape obtained contains the centroid of ${S}$.

5. Let ${V_1,V_2,V_3,V_4}$ be four dimensional linear subspaces in ${\Bbb{R}^8}$ such that the intersection of any two contains only the zero vector. Prove that there exists a linear four dimensional subspace ${W}$ in ${\Bbb{R}^8}$ such that all four vector spaces ${W\cap V_i}$ are two dimensional.

6. Let ${A,B,C}$ be matrices with complex elements such that ${[A,B]=C,\ [B,C]=A}$ and ${[C,A]=B}$, where ${[X,Y]}$ denotes the ${XY-YX}$ commutator of the matrices. Prove that ${e^{4 \pi A}}$ is the identity matrix. [Solution]

7. Let ${\Gamma}$ be a simple curve, lying inside a circle of radius ${r}$, rectifiable and of length ${l}$. Prove that if ${l > kr\pi}$, then there exists a circle of radius ${r}$ which intersects ${\Gamma}$ in at least ${k+1}$ distinct points.

8. For any function ${f: \Bbb{R}^2\rightarrow \Bbb{R}}$ consider the function ${\Phi_f:\Bbb{R}^2\rightarrow [-\infty,\infty]}$ for which ${\Phi_f(x,y)=\limsup_{ z \rightarrow y} f(x,z)}$ for any ${(x,y) \in \Bbb{R}^2}$.

a) Is it true that if ${f}$ is Lebesgue measurable then ${\Phi_f}$ is also Lebesgue measurable?

b) Is it true that if ${f}$ is Borel measurable then ${\Phi_f}$ is also Borel measurable?

9. Let ${D}$ be the complex unit disk ${D=\{x \in \Bbb{C}: |z|<1\}}$, and ${0 a real number. Suppose that ${f:D \rightarrow \Bbb{C}\setminus \{0\}}$ is a holomorphic function such that ${f(a)=1}$ and ${f(-a)=-1}$. Prove that

$\displaystyle \sup_{z \in D} |f(z)| \geq \exp\left(\frac{1-a^2}{4a}\pi\right)$

10. Let ${K}$ be a knot in the ${3}$-dimensional space (that is a differentiable injection of a circle into ${\Bbb{R}^3}$), and ${D}$ be the diagram of the knot (that is the projection of it to a plane so that in exception of the transversal double points it is also an injection of a circle). Let us color the complement of ${D}$ in black and color the diagram ${D}$ in a chessboard pattern, black and white so that zones which share an arc in common are coloured differently. Define ${\Gamma_B(D)}$ the black graph of the diagram, a graph which has vertices in the black areas and every two vertices which correspond to black areas which have a common point have an edge connecting them.

a) Determine all knots having a diagram ${D}$ such that ${\gamma_B(D)}$ has at most ${3}$ spanning trees. (Two knots are not considered distinct if they can be moved into each other with a one parameter set of the injection of the circle)

b) Prove that for any knot and any diagram ${D}$, ${\Gamma_B(D)}$ has an odd number of spanning trees.

11. Let ${X_1,X_2,..}$ be independent random variables with the same distribution, and let ${S_n=X_1+X_2+...+X_n,\ n=1,2,...}$. For what real numbers ${c}$ is the following statement true:

$\displaystyle P\left(\left| \frac{S_{2n}}{2n}- c \right| \leq \left| \frac{S_n}{n}-c\right| \right)\geq \frac{1}{2}\ \ ?$

Miklos Schweitzer competition 2012 (many thanks to Eles Andras for the translation of the problems from Hungarian to English)

## Partition of a set of sets

We partition the $n$-element subsets of an $n^2+n+1$-element set into two classes. Prove that one of the classes contains $n$ many pairwise disjunct sets.
Given a positive integer $n > 3$, prove that the least common multiple of products of the form $x_1x_2...x_k\ (k \geq 1)$ whose factors $x_i$ are positive integers with $x_1+x_2+...+x_n \leq n$ is less than $n!$.


text:
# wing2015

0th

Percentile

##### Snags Segmentation Algorithm

This function is made to be used in lassnags. It implements an algorithms for snags segmentation based on Wing et al (2015) (see references). This is an automated filtering algorithm that utilizes three dimensional neighborhood lidar point-based intensity and density statistics to remove lidar points associated with live trees and retain lidar points associated with snags.

##### Usage
wing2015(neigh_radii = c(1.5, 1, 2), low_int_thrsh = 50,
uppr_int_thrsh = 170, pt_den_req = 3, BBPRthrsh_mat = NULL)
##### Arguments

numeric. A vector of three radii used in quantifying local-area centered neighborhoods. See Wing et al. (2015) reference page 171 and Figure 4. Defaults are 1.5, 1, and 2 for the sphere, small cylinder and large cylinder neighborhoods, respectively.

low_int_thrsh

numeric. The lower intensity threshold filtering value. See Wing et al. (2015) page 171. Default is 50.

uppr_int_thrsh

numeric. The upper intensity threshold filtering value. See Wing et al. (2015) page 171. Default is 170.

pt_den_req

numeric. Point density requirement based on plot-level point density defined classes. See Wing et al. (2015) page 172. Default is 3.

BBPRthrsh_mat

matrix. A 3x4 matrix providing the four average BBPR (branch and bole point ratio) values for each of the three neighborhoods (sphere, small cylinder and large cylinder) to be used for conditional assessments and classification into the following four snag classes: 1) general snag 2) small snag 3) live crown edge snag 4) high canopy cover snag. See Wing et al. (2015) page 172 and Table 2. This matrix must be provided by the user.

##### Details

Note that this algorithm strictly performs a classification based on user input while the original publication's methods also included a segmentation step and some pre- (filtering for first and single returns only) and post-process (filtering for only the snag classified points prior to segmentation) tasks which are now expected to be performed by the user. Also, this implementation may have some differences compared with the original method due to potential mis-interpretation of the Wing et al. manuscript, specifically Table 2 where they present four groups of conditional assessments with their required neighborhood point density and average BBPR values (BBPR = branch and bole point ratio; PDR = point density requirement). This algorithm attributes each point in the point cloud (snagCls column) into the following five snag classes:

• 0: live tree - not a snag

• 1: general snag - the broadest range of snag point situations

• 2: small snag - isolated snags with lower point densities

• 3: live crown edge snag - snags located directly adjacent or intermixing with live trees crowns

• 4: high canopy cover snag - snags protruding above the live canopy in dense conditions (e.g., canopy cover >= 55%).

The current implementation is known to use a large amount of memory for storing the N x k integer matrix returning the near neighbor indices for each point in the point cloud. Improvements are possible in future package versions.

##### References

Wing, Brian M.; Ritchie, Martin W.; Boston, Kevin; Cohen, Warren B.; Olsen, Michael J. 2015. Individual snag detection using neighborhood attribute filtered airborne lidar data. Remote Sensing of Environment. 163: 165-179 https://doi.org/10.1016/j.rse.2015.03.013

• wing2015
##### Examples
# NOT RUN {
LASfile <- system.file("extdata", "MixedConifer.laz", package="lidR")
las <- readLAS(LASfile, select = "xyzi", filter="-keep_first") # Wing also included -keep_single

# For the Wing2015 method, supply a matrix of snag BranchBolePtRatio conditional
# assessment thresholds (see Wing et al. 2015, Table 2, pg. 172)
bbpr_thresholds <- matrix(c(0.80, 0.80, 0.70,
0.85, 0.85, 0.60,
0.80, 0.80, 0.60,
0.90, 0.90, 0.55),
nrow =3, ncol = 4)

# Run snag classification and assign classes to each point
las <- lassnags(las, wing2015(neigh_radii = c(1.5, 1, 2), BBPRthrsh_mat = bbpr_thresholds))

# Plot it all, tree and snag points...
plot(las, color="snagCls", colorPalette = rainbow(5))

# Filter and plot snag points only
snags <- lasfilter(las, snagCls > 0)
plot(snags, color="snagCls", colorPalette = rainbow(5)[-1])

# Wing et al's (2015) methods ended with performing tree segmentation on the
# classified and filtered point cloud using the watershed method

# }

Documentation reproduced from package lidR, version 2.0.0, License: GPL-3

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