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# Biomarker discovery by integrated joint non-negative matrix factorization and pathway signature analyses

## Abstract

Predictive biomarkers are important for selecting appropriate patients for particular treatments. Comprehensive genomic, transcriptomic, and pharmacological data provide clues for understanding relationships between biomarkers and drugs. However, it is still difficult to mine biologically meaningful biomarkers from multi-omics data. Here, we developed an approach for mining multi-omics cell line data by integrating joint non-negative matrix factorization (JNMF) and pathway signature analyses to identify candidate biomarkers. The JNMF detected known associations between biomarkers and drugs such as BRAF mutation with PLX4720 and HER2 amplification with lapatinib. Furthermore, we observed that tumours with both BRAF mutation and MITF activation were more sensitive to BRAF inhibitors compared to tumours with BRAF mutation without MITF activation. Therefore, activation of the BRAF/MITF axis seems to be a more appropriate biomarker for predicting the efficacy of a BRAF inhibitor than the conventional biomarker of BRAF mutation alone. Our biomarker discovery scheme represents an integration of JNMF multi-omics clustering and multi-layer interpretation based on pathway gene signature analyses. This approach is also expected to be useful for establishing drug development strategies, identifying pharmacodynamic biomarkers, in mode of action analysis, as well as for mining drug response data in a clinical setting.

## Introduction

Precision medicine for cancer patients with molecular targeted drugs and predictive biomarkers is expected to lead to a paradigm shift from one-size-fits-all medicine to patient-specific medicine1. In particular, in the era of cancer immunology, the immunohistochemistry of programmed death ligand-1 (PD-L1) expression levels has been approved as a companion diagnostic for the anti-PD-1 antibody pembrolizumab; however, the identification of useful biomarkers remains a significant issue2. Selecting appropriate patients for a particular treatment using predictive biomarkers will certainly help to increase therapeutic effectiveness and reduce toxicities. Thus, it is important to identify reliable predictive biomarkers to select the right patient for the right drug.

Comprehensive genomic and pharmacological data of large collections of cancer cell lines have been published as the Cancer Cell Line Encyclopedia (CCLE)3,4. These cell line databases provide mutation, copy number alteration, and mRNA expression profiles, as well as the results of tests of the sensitivities of cells to growth inhibition induced by various compounds or drugs. Although there is some discordance between databases, especially in terms of the compound sensitivity profiles, these databases generally show reasonable consistency5,6.

These multi-dimensional genomic and pharmacological datasets have been used to perform multi-omics analyses with the goal of understanding the relationships between cancer genomes and drug responders. The NCI DREAM challenge is an example of an approach leading to remarkable improvements in this area7, in which several prediction models were proposed to estimate sensitivity to compounds based on genetic information. The top-performing method was found to be a kernel method with multiview and multitask learning, which uses all of the genetic profiles provided7. Although this challenge is focused on providing a benchmarked set of algorithms, it is difficult to translate the results obtained from the predictors for clinical application. This is because the models simultaneously require genetic, epigenomic, and proteomic data, and such comprehensive models make for challenging biological interpretations. Moreover, other research programs have focused on discovering useful biomarker candidates in clinical settings rather than relying on predictive performance. Several well-known biomarker and drug associations were detected using analysis of variance (ANOVA)- or regression-based analyses from multi-omics data3,4. However, some biological features such as tissue-specific expression are correlated, which poses a limitation in the reliability of ANOVA with multiplicity and regression analysis with multicollinearity in handling these features. Furthermore, the most common type of predictive biomarker measured clinically with a companion diagnosis kit is a single gene that is equivalent to the therapeutic target itself or a gene that is biologically relevant to the target8. Therefore, it is still challenging to efficiently utilize comprehensive genomic data to determine an appropriate treatment strategy. In this study, we sought to resolve these issues to facilitate the use of multi-omics analyses for understanding relationships between the cancer genome and drug responders through development of comprehensive prediction models with multi-genetic features. Since the choice of predictive biomarkers for suggesting treatment options to patients should be based on a biological rationale, we focused on detecting biologically meaningful biomarkers rather than merely developing comprehensive multi-omics predictors.

Non-negative matrix factorization (NMF) is an unsupervised approach that can highlight outliers or extreme characteristics in a non-negative input matrix X according to its parts-based representation nature9. Matrix X is then factorized by the non-negative submatrices W and H. NMF has emerged as one of the most useful algorithms currently available in the cancer genome research field. An NMF method was used to generate mutational signatures for 96 trinucleotide mutation patterns from the genomes of cancer patients10,11. For example, this method detected that smoking-related C > A mutations at NpCpN and POLE mutations were related to C > A and T > G mutations in a TpCpT and TpTpT context, respectively. Thus, a mutational signature or NMF approach can help to decompose the multiple effects of a carcinogen based on a patient’s combined mutation pattern. Furthermore, NMF can classify tumour subtypes from microarray data. Indeed, NMF was used to identify a small number of gene combinations (metagenes) whose profiles represent features that can distinguish among leukaemia and lung cancer subtypes12. NMF is essentially applicable for a single-input matrix such as face imaging pixels and mutational signature profiles, although it has potential to be further extended to multiple inputs. Joint non-negative matrix factorization (JNMF) is fitted for multiple inputs with the same row size, which generates a common sub matrix W and individual sub matrices Hs13. JNMF can therefore be used to detect common clusters (co-modules) from mRNA expression, microRNA expression, and DNA methylation data of cancer patients. Thus, we hypothesized that JNMF would be a suitable method to handle several multi-omics datasets simultaneously. Moreover, among the many techniques available to handle multiple inputs14, JNMF is theoretically and practically equivalent to a standard NMF method with concatenated inputs.

With respect to resolving the biological interpretation challenge, pathway analysis can be a useful tool to annotate a given set of genes in a biologically meaningful manner15. Ingenuity Pathway Analysis (IPA) is one of the most beneficial tools available to understand the association of various types of molecules, and provides causal networks based on biological relationships curated from the literature16. Furthermore, gene signature analysis enables inferences on pathway activation and dependency17,18. Therefore, pathway and gene signature analyses are effective for understanding particular gene sets identified as NMF co-modules.

Our research objective was to identify promising candidate biomarkers using multi-dimensional genomic and pharmacological data from a collection of cell lines. The approach used is based on the integration of JNMF multi-omics clustering with multi-layer interpretation based on pathway gene signature analyses (Fig. 1). This scheme enabled us to identify novel rationale-based biomarkers as well as known clinically validated biomarkers.

## Results

### JNMF with missing data

To illustrate the robustness of our multi-omics clustering method against missing values, JNMF was first applied to simulated data. Three simulated input matrices were generated: a simulated compound sensitivity matrix, simulated mutation matrix, and simulated mRNA expression matrix. Four co-modules were predefined in these simulated matrices, in line with a previous report13. To best mimic real data, the simulated genetic mutation matrix was represented in a binary format, whereas the simulated compound sensitivity and mRNA expression matrices were represented in a continuous format. The simulated compound sensitivity matrix contained certain missing values randomly (10%), because typical sensitivity metrics (IC 50 , GI50, AUC, etc.) often fail due to the lack of measurement of a compound in a cell line or after filtering out noisy results. The simulated matrices of mutation and expression profiles contained missing rows randomly (10%), since public genomic and transcriptomic data cannot always be fully assigned to all cell lines for compound sensitivity data.

JNMF for the three simulated matrices with factorization rank k = 4 correctly revealed the four predefined clusters (Fig. 2). JNMF with X1, X2, and X3 inputs returned H1, H2, H3, and W submatrices. WH1 clearly reproduced X1, resulting in four modules (blue submatrices), and WH2 and WH3 contained three modules. Therefore, JNMF detected co-clusters hidden in the input matrices despite the presence of missing values. The dimensionality reduction effect of JNMF also results in a noise reduction effect. Furthermore, JNMF could interpolate the missing values. This property of JNMF might enable predicting the mutation status from expression profiles, as well as predicting compound sensitivity from genomic profiles. However, further experimental validation must be conducted to validate these applications.

We next investigated the effects of missing and noise values on the results of JNMF. For the same simulated datasets, the following four parameters were systematically examined by a 0.1 grid to check the consistency of elements in the four co-modules: m as the missing rate of all input Xs, and a, b, and c as the noise rates of X1, X2, and X3, respectively. Regression analysis revealed that the consistency of the co-module in W, y W , is affected by the parameters m, a, and c (Fig. S1a), whereas the other consistency indexes y H1 , y H2 , and y H3 are affected by {m, a}, {m, b}, and {m, c}, respectively (Fig. S1b–d). In this artificial case, a missing ratio m = 0.1 had a slight influence on the results.

### Detection of known biomarkers by JNMF

To discover the relationship between the compound sensitivity and multi-omics profiles of cell lines, we conducted JNMF with factorization rank k = 40 for the CCLE data set3 as follows: compound sensitivity X 1 , mutation X 2 , copy number amplification X 3 , copy number loss X 4 , mRNA expression X 5 , and tumour type X 6 (Table S1). JNMF was applied to find 40 meta-profiles for cell lines as a W matrix and 40 co-modules for genetic and pharmacological features as H matrices simultaneously. We repeated 10 trials of JNMF with random initial values. The results showed that the objective function sufficiently converged in all 10 trials (Fig. S2). Furthermore, a consensus matrix for W showed high concordance between trials with a cophenetic correlation coefficient of 0.91 (Fig. S3). This consensus matrix contained robust clusters with melanoma, blood cancer, and hypermutated profiles. This consistency reflects the fact that similar JNMF clusters could be obtained for any initial condition. Thus, JNMF would be able to detect unique characteristics from multi-dimensional genomic and pharmacologic data. The best result was selected as that showing the smallest objective function value among the 10 trials, which was used for further analysis.

JNMF revealed several well-established relationships between drugs and biomarkers. For example, BRAF-mutated melanoma cells and patients in preclinical and clinical trials show specific sensitivity to the BRAF inhibitors PLX4720 and its structural analogue, PLX4032 (vemurafenib)19,20. In line with this knowledge, our JNMF result returned a co-module (#12) that contained PLX4720, BRAF mutation, and melanoma tumour type (Fig. 3a). Another co-module (#5) was related to human epidermal growth factor receptor 2 (HER2)-activated breast tumours, which was enriched with the features of HER2 amplification, overexpression, and breast tumour, and also showed sensitivity to the HER2 inhibitor lapatinib (Fig. 3b). Ultimately, we selected a factorization rank of 40 based on high consistency of the JNMF results with biologically useful knowledge. Furthermore, each genetic or pharmacological feature belonged to one co-module according to the consensus and connectivity matrices scheme. In addition to the HER2-breast cancer (#5) and BRAF–melanoma (#12) co-modules, other cancer-related co-modules were selected, such as those representing the hypermutated phenotype (#3) and responders to receptor tyrosine kinase inhibitors (#28) (Table S2).

### Pathway analysis predicted activation of microphthalmia-associated transcription factor (MITF) in the melanoma cluster

To interpret the biological relationships in the obtained multi-omics JNMF co-module clusters, pathway analysis was performed for the most highly expressed gene sets in the co-modules to investigate whether or not a specific pathway was activated. IPA upstream analysis estimated module-specific upstream regulators, including transcription factors (Table S2), and suggested that the transcription factor MITF was activated in co-module #12, which was highly significant among co-modules (IPA overlap p-value = 6.91E-39, Fig. S4). Therefore, we focused on co-module #12, which was sensitive to BRAF inhibition and included strong melanoma characteristics, and was also one of the highly reproducible modules among the 10 trials as mentioned above. Furthermore, IPA protein-protein interaction (PPI) analysis for co-module #12 showed that melanoma-related transcription factors, such as MITF, PAX3, SOX8, and SOX10, formed a sub-network (Fig. S5). Thus, these analyses of upstream regulator estimation and PPI network elucidated MITF activation. Since co-module #12 also included MITF amplification and overexpression of MITF itself, it was presumed that the gene sets were highly expressed as a result of MITF activation associated with MITF amplification. Therefore, it seems that MITF amplification and activation is a characteristic of this module.

### MITF activation signature as a novel predictive biomarker for BRAF inhibition

We further examined the input data to explore the utility of MITF amplification and activation as a biomarker in addition to BRAF mutation status for predicting the response to a BRAF inhibitor. First, we detected relationships between the sensitivity profile to a BRAF inhibitor and genetic features, including BRAF mutation, MITF amplification, and expression profiles of MITF target genes, present in the JNMF input set (Fig. 4). Next, a MITF activation signature was generated based on the genes regulated by MITF in co-module #12, and the correlation between the MITF activation score and PLX4720 sensitivity profile in the JNMF input set was confirmed (rank correlation = 0.17, p-value = 0.0004, Fig. 4). This evidence suggests that JNMF can detect reasonable relationships hidden in multi-omics input data. Finally, we found that the activation status of MITF alone could serve as a useful novel biomarker to indicate the sensitivity to a BRAF inhibitor. Although a correlation between BRAF mutation and MITF activation was detected, they did not completely overlap in melanoma cells. However, tumours with both BRAF mutation and MITF activation were more sensitive to PLX4720 compared to tumours with BRAF mutation but without MITF activation (Fig. 5).

### Drug differentiation strategy using JNMF

Similar to biomarker discovery, an essential component of drug development is to understand the unique characteristics of individual drugs. Utilizing the parts-based representation of NMF can help to reveal important differences among drugs based on the results of JNMF.

Indeed, we detected remarkable differences in the JNMF results between two compounds with similar overall drug sensitivity profiles. For example, the RAS/BRAF/MEK signal transduction pathway is known to play an important role in tumour development in multiple cancer types. Thus, both BRAF and MEK inhibitors show similar biological effects on cancer cells. However, JNMF found that co-module #30, enriched with the features of colorectal tumours, was only sensitive to MEK inhibitor and did not exert efficacy to BRAF inhibitor (Fig. 3a). Both epidermal growth factor receptor (EGFR) and HER2 belong to the ERBB protein family. Thus, a single ERBB inhibitor has been developed for treating EGFR-mutated or HER2-amplified cancers. However, in the present analysis, co-module #5 showed that HER2-amplified and overexpressed breast cancers were sensitive to HER2 inhibition but were not sensitive to EGFR inhibition (Fig. 3b). Given that the present JNMF-based approach could detect critical, biologically verified differences in drugs, even between drugs with similar overall sensitivity profiles, it shows good promise for developing unique biomarker-based strategies to design novel compounds, with more reliability than currently available approaches.

## Discussion

The proposed JNMF method adapted to handle missing data allows for the integration of multiple genomic and pharmacological datasets, and our multi-dimensional clustering approach of JNMF could efficiently extract genetic alterations related to sensitivity profiles for specific compounds. Moreover, using simulated data, JNMF correctly detected co-modules or multi-dimensional clusters with predefined conditions containing 10% missing values.

By exploiting the parts-based representation of NMF, we successfully excluded the influence of other interacting factors to extract melanoma-specific features. Further, by combining known biological knowledge via pathway analysis, MITF downstream genes emerged as candidate biomarkers that can be interpreted based on a biological rationale rather than from simple prediction analysis derived from combining the influence of multiple features. Furthermore, use of a gene signature approach enabled estimation of the activation levels of the transcriptional regulator MITF and could predict responders and non-responders to BRAF inhibitors by integrating information of the known and candidate biomarkers. Specifically, tumours with both BRAF mutation and MITF activation were more sensitive to PLX4720 than tumours with BRAF mutation without MITF activation (Fig. 5). Therefore, activation of the BRAF/MITF axis might be a more appropriate biomarker for predicting the efficacy of a BRAF inhibitor than BRAF mutation alone, which is frequently used as a predictive biomarker in preclinical studies and clinical trials (Fig. S6).

Several studies have reported that MITF plays important roles in cell cycle progression by activating downstream genes in melanoma21,22. Consistent with our findings, forced expression of MITF conferred melanoma cells with sensitivity to BRAF/MEK inhibitors23. However, there is also a conflicting report that MITF activation might be involved in the mechanism of resistance to BRAF inhibition therapy24. Thus, our candidate biomarker should be evaluated in additional preclinical studies as well as in clinical trials.

With respect to lapatinib, JNMF clearly identified two major responder groups, breast cancer and lung cancer cells. This result is supported by the fact that lapatinib has been approved for use in clinical settings for treating HER2-positive breast cancer, and was shown to be effective in some types of lung cancer in a preclinical experiment25. Nevertheless, traditional chemotherapeutic agents did not show a peak in any of the co-modules since the majority of the included cell lines are sensitive to these agents. Therefore, it is still difficult to determine the characteristics of responder cell lines to broadly active compounds based on genomic and transcriptomic information.

Elastic net regression analysis was previously shown to effectively predict drug sensitivity based on several genetic features3. However, this approach selects one representative feature among highly correlated features, which makes it difficult to select the correlated MITF target genes simultaneously (Fig. 3). In the CCLE report, although the BRAF mutation emerged as a strong predictor of response to the PLX4720 compound, the MITF target genes and MITF amplification (except GAPDHS expression) were not selected as significant predictors, which is possibly because these features are sufficiently correlated with BRAF mutation.

As represented by the NCI DREAM challenge, drug sensitivity profiles are typically predicted using comprehensive multi-omics data with incorporation of available biological knowledge. However, there is still a substantial gap between such basic research and companion diagnostic development through clinical trials. One possible reason for this limitation is that a predictive model must be sufficiently interpreted and explained to both the patients and doctors when used for patient selection in a clinical setting. Thus, a future challenge might be how to best explain a diagnostic result derived from multi-omics biomarkers in a clinical situation. Accordingly, in addition to refining multi-omics itself, another crucial task for practical realization is to establish a user-friendly interpretable format for outputting the data.

Prediction of missing values by JNMF is expected to be a useful feature for inferring a mutation status or drug sensitivity profile from other types of multi-omics information such as expression profiles. Given that our JNMF approach could efficiently predict missing values from simulated data, and the clustering was useful for detecting co-modules for datasets containing missing values, it is necessary to further verify whether this missing value prediction using JNMF is applicable to actual data.

There are several successful examples of drug development using a biomarker-based strategy, including the development of third-generation EGFR inhibitors and PARP inhibitors. Molecular and clinical profiles of the investigational drugs themselves were similar with respect to the mode-of-action (MOA). However, the patient stratification strategy for clinical trials based on the biomarker differed for different drugs, signifying the importance of a useful predictive biomarker for successful drug development. The proposed approach of incorporating the JNMF method with biomarker selection could help to increase the success rate of drug development by providing a differentiated and superior biomarker strategy compared with those used previously for investigational drugs with a similar MOA.

Overall, we have presented an NMF-based unsupervised approach for discovering useful biomarkers using in vitro multi-dimensional cell line data. The generated JNMF co-modules are based on meta-profiles derived from whole genomic and pharmacological data. Therefore, candidate biomarkers are derived from the relationship between multi-therapeutic targets and multi-genetic alterations. This concept is quite different from a single therapeutic model of a compound such as ANOVA and elastic net regression. Our JNMF simultaneously detects differences between compound profiles and should thus be useful for exploring drug development strategies while revealing genomic characteristics or therapeutic targets for unmet medical needs. Furthermore, the proposed biomarker discovery scheme will be useful for finding pharmacodynamic biomarkers and for MOA analysis at the preclinical stage. Together, this approach and similar developments should accelerate translational research and mining for clinical significance using clinical response data.

## Methods

### Data summarization, normalization, and preprocessing

Mutation, copy number alterations, mRNA expression, and compound sensitivity profiles were collected from the CCLE (Table S1). Mutation profiles were converted into binary data of mutant type (1) or wild type (0) for each gene. Copy number profiles generated by the GISTIC algorithm26 were divided into amplification (GISTIC score: +2) and deletion (GISTIC score: −2) profiles to obtain respective binary datasets of amplification and deletion. mRNA expression profiles were normalized from the log2 intensity values to a 0–1 score for each gene. The concentration at which the drug response reached an absolute inhibition of 50% (IC50) was used as an index of the compound sensitivity. The IC50 values were converted to a score of insensitive (0) and sensitive (1) according to the following formula:

$${\rm{Normalized}}\,{{\rm{IC}}}_{50}=\frac{{x}_{max}-x}{{x}_{max}-{x}_{min}},$$
(1)

where x indicates the IC50 value and x max and x min are the maximum and minimum IC50 values of a compound among the CCLE cell lines tested.

A tumour-type binary matrix was also obtained based on the CCLE lineage information. Thus, a total of six 0–1-normalized matrices X1, X2, …, X6 were obtained: compound sensitivity, mutation, amplification, copy number deletion, mRNA expression, and tumour type. A total of 504 cell lines with sensitivity data for at least one compound were subjected to subsequent analysis.

### JNMF and mask matrix

JNMF was performed for N matrices. The objective function of JNMF is a squared Euclidean error function and formulated as

$${\rm{\min }}\,\sum _{I=1}^{N}{\Vert {X}_{I}-W{H}_{I}\Vert }_{F}^{2}$$
(2)

where X I is an m × n I input matrix, W, H I represents m × k, k × n I output matrices, k is the factorization rank of JNMF, and F shows Frobenius norm. N = 3 was used for simulated data and N = 6 matrices from CCLE data were obtained. Then, the standard multiplicative update rules were selected for NMF calculation as follows:

$${W}_{ia}\leftarrow {W}_{ia}\frac{{({\sum }_{J=1}^{N}({X}_{J}{H}_{J}^{T}))}_{ia}}{{(W{\sum }_{K=1}^{N}({H}_{K}{H}_{K}^{T}))}_{ia}}$$
(3)
$${({H}_{I})}_{a\mu }\leftarrow {({H}_{I})}_{a\mu }\frac{{({W}^{T}{X}_{I})}_{a\mu }}{{({W}^{T}W{H}_{I})}_{a\mu }},\,I=1,\cdots ,N.$$
(4)

The NMF procedure was modified to handle matrices containing missing values using a mask matrix. In brief, we used a weighted NMF approach27. The mask matrix M has the same dimensions as the input matrix X, with 1 in the case where there is a value in X in each matrix element and 0 in the case where there is not. For the execution of JNMF, the product of each matrix element of X and M is obtained so that the normal JNMF calculation is executed when the value of X exists and is ignored when the value of X does not exist.

$${\rm{\min }}\,\sum _{I=1}^{N}{\Vert {M}_{I}\,\circ \,({X}_{I}-W{H}_{I})\Vert }_{F}^{2}$$
(5)
$${W}_{ia}\leftarrow {W}_{ia}\frac{{({\sum }_{J=1}^{N}(({M}_{J}\circ {X}_{J}){H}_{J}^{T}))}_{ia}}{{({\sum }_{K=1}^{N}({M}_{K}\circ (W{H}_{K}){H}_{K}^{T}))}_{ia}}$$
(6)
$${({H}_{I})}_{a\mu }\leftarrow {({H}_{I})}_{a\mu }\frac{{({W}^{T}({M}_{I}\circ {X}_{I}))}_{a\mu }}{{({W}^{T}({M}_{I}\circ (W{H}_{I})))}_{a\mu }},\,I=1,\cdots ,\,N,$$
(7)

where $$A\,\circ \,B=[{a}_{ij}{b}_{ij}]$$ represents the Hadamard product.

This JNMF update procedure for W and each H I is executed at 5,000 iterations with an appropriate factorization rank k, and a convergence is observed (Fig. S2). It is repeated for a prescribed number of times $$T$$ and a result is selected in which the difference between X and WH is minimal among the T results. In addition, a consensus matrix and its cophenetic correlation coefficient from all T results were obtained to examine the reproducibility.

### Pathway and gene signature analyses

IPA16 was used to predict the activated pathways in JNMF co-modules. IPA upstream and IPA protein-protein interaction analyses are used to elucidate the common regulators of gene sets in the JNMF co-module. For a gene set (n g genes), the average expression level was taken as the gene signature representing pathway activation according to the following formula:

$${\rm{Signature}}\,{{\rm{Score}}}_{i}=\,\frac{1}{{n}_{g}}\sum _{j}^{{n}_{g}}(\frac{{x}_{ij}-\mathop{{\rm{\min }}}\limits_{i}{x}_{ij}}{\mathop{{\rm{\max }}}\limits_{i}{x}_{ij}-\mathop{{\rm{\min }}}\limits_{i}{x}_{ij}})$$
(8)

where x ij indicates the mRNA expression level of a gene j in a cell i.

Signature score is calculated for n g genes in mRNA profiles X5, obtained by integrating the JNMF co-module and IPA knowledge.

### Data availability

The data that support the findings of this study are available on the Cancer Cell Line Encyclopedia (https://portals.broadinstitute.org/ccle)3 and the cBioPortal for Cancer Genomics (http://www.cbioportal.org/)28,29.

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## Acknowledgements

Computational resources were provided by the supercomputer system at the Human Genome Center, the Institute of Medical Science, the University of Tokyo.

## Author information

Authors

### Contributions

N.F. designed and conducted the data analyses. N.F., S.M. and K.M. wrote the manuscript. K.N. conceived of and designed the study. All authors read and approved the final manuscript.

### Corresponding author

Correspondence to Kenta Nakai.

## Ethics declarations

### Competing Interests

N.F. and S.M. are employees of Taiho Pharmaceutical Co., Ltd. (Tokyo, Japan).

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Fujita, N., Mizuarai, S., Murakami, K. et al. Biomarker discovery by integrated joint non-negative matrix factorization and pathway signature analyses. Sci Rep 8, 9743 (2018). https://doi.org/10.1038/s41598-018-28066-w

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## scalable Langevin exact algorithm [armchair Read Paper]

Posted in Books, pictures, Statistics, Travel, University life with tags , , , , , , , , , , , on June 26, 2020 by xi'an

So, Murray Pollock, Paul Fearnhead, Adam M. Johansen and Gareth O. Roberts presented their Read Paper with discussions on the Wednesday aft! With a well-sized if virtual audience of nearly a hundred people. Here are a few notes scribbled during the Readings. And attempts at keeping the traditional structure of the meeting alive.

In their introduction, they gave the intuition of a quasi-stationary chain as the probability to be in A at time t while still alice as π(A) x exp(-λt) for a fixed killing rate λ. The concept is quite fascinating if less straightforward than stationarity! The presentation put the stress on the available recourse to an unbiased estimator of the κ rate whose initialisation scaled as O(n) but allowed a subsampling cost reduction afterwards. With a subsampling rat connected with Bayesian asymptotics, namely on how quickly the posterior concentrates. Unfortunately, this makes the practical construction harder, since n is finite and the concentration rate is unknown (although a default guess should be √n). I wondered if the link with self-avoiding random walks was more than historical.

The initialisation of the method remains a challenge in complex environments. And hence one may wonder if and how better it does when compared with SMC. Furthermore, while the motivation for using a Brownian motion stems from the practical side, this simulation does not account for the target π. This completely blind excursion sounds worse than simulating from the prior in other settings.

One early illustration for quasi stationarity was based on an hypothetical distribution of lions and wandering (Brownian) antelopes. I found that the associated concept of soft killing was not necessarily well received by …. the antelopes!

As it happens, my friend and coauthor Natesh Pillai was the first discussant! I did no not get the details of his first bimodal example. But he addressed my earlier question about how large the running time T should be. Since the computational cost should be exploding with T. He also drew a analogy with improper posteriors as to wonder about the availability of convergence assessment.

And my friend and coauthor Nicolas Chopin was the second discussant! Starting with a request to… leave the Pima Indians (model)  alone!! But also getting into a deeper assessment of the alternative use of SMCs.

## scalable Langevin exact algorithm [Read Paper]

Posted in Books, pictures, Statistics, University life with tags , , , , , , , , , , , , , , , , on June 23, 2020 by xi'an

Murray Pollock, Paul Fearnhead, Adam M. Johansen and Gareth O. Roberts (CoI: all with whom I have strong professional and personal connections!) have a Read Paper discussion happening tomorrow [under relaxed lockdown conditions in the UK, except for the absurd quatorzine on all travelers|, but still in a virtual format] that we discussed together [from our respective homes] at Paris Dauphine. And which I already discussed on this blog when it first came out.

Here are quotes I spotted during this virtual Dauphine discussion but we did not come up with enough material to build a significant discussion, although wondering at the potential for solving the O(n) bottleneck, handling doubly intractable cases like the Ising model. And noticing the nice features of the log target being estimable by unbiased estimators. And of using control variates, for once well-justified in a non-trivial environment.

“However, in practice this simple idea is unlikely to work. We can see this most clearly with the rejection sampler, as the probability of survival will decrease exponentially with t—and thus the rejection probability will often be prohibitively large.”

“This can be viewed as a rejection sampler to simulate from μ(x,t), the distribution of the Brownian motion at time  t conditional on its surviving to time t. Any realization that has been killed is ‘rejected’ and a realization that is not killed is a draw from μ(x,t). It is easy to construct an importance sampling version of this rejection sampler.”

## is there such a thing as optimal subsampling?

Posted in Books, Statistics, University life with tags , , , , , , , , , , , , , on June 12, 2020 by xi'an

This idea of optimal thinnin and burnin has been around since the early days of the MCMC revolution and did not come up with a definite answer. For instance, from a pure estimation perspective, subsampling always increases the variance of the resulting estimator. My personal approach is to ignore both burnin and thinnin and rather waste time on running several copies of the code to check for potential discrepancies and get a crude notion of the variability. And to refuse to answer to questions like is 5000 iterations long enough for burnin?

A recent arXival by Riabiz et al. readdresses the issue. In particular concerning this notion that the variance of the subsampled version is higher: this only applies to a deterministic subsampling, as opposed to an MCMC-based subsampling (although this intricacy only makes the problem harder!). I however fail to understand the argument in favour of subsampling based on storage issues (p.4), as a dynamic storage of the running mean for all quantities of interest does not cost anything if the integrand is not particularly demanding. I also disagree at the pessimistic view that the asymptotic variance of the MCMC estimate is hard to estimate: papers by Flegal, Hobert, Jones, Vat and others have rather clearly shown how batch means can produce converging estimates of this asymptotic variance.

“We do not to attempt to solve a continuous optimisation problem for selection of the next point [in the sample]. Such optimisation problems are fundamentally difficult and can at best be approximately solved. Instead, we exactly solve the discrete optimisation problem of selecting a suitable element from a supplied MCMC output.”

One definitely positive aspect of the paper is that the (thinning) method is called Stein thinning, in connection with Stein’s discrepancy, and this honours Charles Stein. The method looks at the optimal subsample, with optimality defined in terms of minimising Stein’s discrepancy from the true target over a reproducible kernel Hilbert space. And then over a subsample to minimise the distance from the empirical distribution to the theoretical distribution. The kernel (11) is based on the gradient of the target log density and the solution is determined by greedy algorithms that determine which next entry to add to the empirical distribution. Which is of complexity O(nm2) if the subsample is of size m. Some entries may appear more than once and the burnin step could be automatically included as (relatively) unlikely values are never selected (at least this was my heuristic understanding). While the theoretical backup for the construct is present and backed by earlier papers of some of the authors, I do wonder at the use of the most rudimentary representation of an approximation to the target when smoother versions could have been chosen and optimised on the same ground. And I am also surprised at the dependence of both estimators and discrepancies on the choice of the (sort-of) covariance matrix in the inner kernel, as the ODE examples provided in the paper (see, e.g., Figure 7). (As an aside and at a shallow level, the approach also reminded me of the principal points of my late friend Bernhard Flury…) Storage of all MCMC simulations for a later post-processing is of course costly in terms of storage, at O(nm). Unless a “secretary problem” approach can be proposed to get sequential. Another possible alternate would be to consider directly the chain of the accepted values (à la vanilla Rao-Blackwellisation). Overall, since the stopping criterion is based on a fixed sample size, and hence depends on the sub-efficiency of evaluating the mass of different modes, I am unsure the method is anything but what-you-get-is-what-you-see, i.e. prone to get misled by a poor exploration of the complete support of the target.

“This paper focuses on nonuniform subsampling and shows that it is more efficiency than uniform subsampling.”

Two weeks later, Guanyu Hu and Hai Ying Wang arXived their Most Likely Optimal Subsampled Markov Chain Monte Carlo, in what I first thought as an answer to the above! But both actually have little in common as this second paper considers subsampling on the data, rather than the MCMC output, towards producing scalable algorithms. Building upon Bardenet et al. (2014) and Korattikara et al. (2014).  Replacing thus the log-likelihood with a random sub-sampled version and deriving the sample size based on a large deviation inequality. By a Cauchy-Schwartz inequality, the authors find sampling probabilities proportional to the individual log-likelihooods. Which depend on the running value of the MCMC’ed parameters. And thus replaced with the values at a fixed parameter, with cost O(n) but only once, but no so much optimal. (The large deviation inequality therein is only concerned with an approximation to the log-likelihood, without examining the long term impact on the convergence of the approximate Markov chain as this is no longer pseudo-marginal MCMC. For instance, both current and prospective log-likelihoods are re-estimated at each iteration. The paper compares with uniform sampling on toy examples,  to demonstrate a smaller estimation error for the statistical problem, rather than convergence to the true posterior.)

## simulating hazard

Posted in Books, Kids, pictures, Statistics, Travel with tags , , , , , , , , , , , , on May 26, 2020 by xi'an

A rather straightforward X validated question that however leads to an interesting simulation question: when given the hazard function h(·), rather than the probability density f(·), how does one simulate this distribution? Mathematically h(·) identifies the probability distribution as much as f(·),

$1-F(x)=\exp\left\{ \int_{-\infty}^x h(t)\,\text{d}t \right\}=\exp\{H(x)\}$

which means cdf inversion could be implemented in principle. But in practice, assuming the integral is intractable, what would an exact solution look like? Including MCMC versions exploiting one fixed point representation or the other.. Since

$f(x)=h(x)\,\exp\left\{ \int_{-\infty}^x h(t)\,\text{d}t \right\}$

using an unbiased estimator of the exponential term in a pseudo-marginal algorithm would work. And getting an unbiased estimator of the exponential term can be done by Glynn & Rhee debiasing. But this is rather costly… Having Devroye’s book under my nose [at my home desk] should however have driven me earlier to the obvious solution to… simply open it!!! A whole section (VI.2) is indeed dedicated to simulations when the distribution is given by the hazard rate. (Which made me realise this problem is related with PDMPs in that thinning and composition tricks are common to both.) Besides the inversion method, ie X=H⁻¹(U), Devroye suggests thinning a Poisson process when h(·) is bounded by a manageable g(·). Or a generic dynamic thinning approach that converges when h(·) is non-increasing.

## generalised Poisson difference autoregressive processes

Posted in pictures, Statistics, Travel, University life with tags , , , , , , , , on February 14, 2020 by xi'an

Yesterday, Giulia Carallo arXived the paper on generalised Poisson difference autoregressive processes that is a component of her Ph.D. thesis at Ca’ Foscari Universita di Venezia and to which I contributed while visiting Venezia last Spring. The stochastic process under study is integer valued as a difference of two generalised Poisson variates, made dependent by an INGARCH process that expresses the mean as a regression over past values of the process and past means. Which can be easily simulated as a difference of (correlated) Poisson variates. These two variates can in their turn be (re)defined through a thinning operator that I find most compelling, namely as a sum of Poisson variates with a number of terms being a (quasi-) Binomial variate depending on the previous value. This representation proves useful in establishing stationarity conditions on the process. Beyond establishing various properties of the process, the paper also examines how to conduct Bayesian inference in this context, with specialised Gibbs samplers in action. And comparing models on real datasets via Geyer‘s (1994) logistic approximation to Bayes factors.

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# elefanten berliner tierpark

If u is twice differentiable then integration by parts yields (2.2) or, equivalently, (2.3) div (a(\i'u)) = 0 This partial differential equation is known as the minimal surface equation. Tobias Holck Colding and William P. Minicozzi, II. However, the term is used for more general surfaces that may self-intersect or do not have constraints. A simpler version of the equation is obtained by lineariza-tion: we assume that |Du|2 ˝ 1 and neglect it in the denominator. We give a counterexample in R 2. [5], Minimal surfaces have become an area of intense scientific study, especially in the areas of molecular engineering and materials science, due to their anticipated applications in self-assembly of complex materials. 303 0 obj <>/Filter/FlateDecode/ID[<9905AF4C536B704FAAAE36E66E929823>]/Index[189 129]/Info 188 0 R/Length 287/Prev 1231586/Root 190 0 R/Size 318/Type/XRef/W[1 2 1]>>stream In discrete differential geometry discrete minimal surfaces are studied: simplicial complexes of triangles that minimize their area under small perturbations of their vertex positions. This property establishes a connection with soap films; a soap film deformed to have a wire frame as boundary will minimize area. The thin membrane that spans the wire boundary is a minimal surface; of all possible surfaces that span the boundary, it is the one with minimal energy. [7] In contrast to the event horizon, they represent a curvature-based approach to understanding black hole boundaries. So we get the minimal surface equation (MSE): div(ru p 1 + jruj2) We call the solution to this equation is minimal surface. My question is the following: since a geodesic is just a special case of a minimal surface, is there some analogous equation for the deviation vector field between two "infinitesimally nearby" minimal (or more generally, extremal) surfaces? In Fig. If the projected Gauss map obeys the Cauchy–Riemann equations then either the trace vanishes or every point of M is umbilic, in which case it is a piece of a sphere. Expanding the minimal surface equation, and multiplying through by the factor (1 + jgrad(f)j2)3=2 weobtaintheequation (1 + f2 y)f xx+ (1 + f 2 x)f yy 2f xf yf xy= 0 BIFURCATION FOR MINIMAL SURFACE EQUATION IN HYPERBOLIC 3-MANIFOLDS ZHENG HUANG, MARCELLO LUCIA, AND GABRIELLA TARANTELLO Abstract. A famous example is the Olympiapark in Münich by Frei Otto, inspired by soap surfaces. Show that the Euler{Lagrange equation for E[v] = Z 1 2 jrvj 2 vf dx (v : !R) is Poisson’s equation u = f: Problem 2. The local least area and variational definitions allow extending minimal surfaces to other Riemannian manifolds than R3. endstream endobj startxref 1 = 0 from the minimal surface equation Lf= 1 + f2 2 f 11 2f 1f 2f 12 + 1 + f2 1 f 22 = 0: Bernstein™s way of computation is take derivative of the equation with respect to x 1 and eliminate the f 22 term in the resulting equation by the equation: 1 + f2 2 f 111 2f 1f 2f 121+ 1 + f2 1 f 221+2f 2f 21f 11! He derived the Euler–Lagrange equation for the solution. An interior gradient bound for classical solutions of the minimal surface equation in n variables was established by Bombieri, De Giorgi, and Miranda in 1968. Savans, 10:477–510, 1785. etY another equivalent statement is that the surface is Minimal if and only if it's principal curvatures are equal in … By contrast, a spherical soap bubble encloses a region which has a different pressure from the exterior region, and as such does not have zero mean curvature. In 1776 Jean Baptiste Marie Meusnier discovered that the helicoid and catenoid satisfy the equation and that the differential expression corresponds to twice the mean curvature of the surface, concluding that surfaces with zero mean curvature are area-minimizing. A minimal surface is a surface each point of which has a neighborhood that is a surface of minimal area among the surfaces with the same boundary as the boundary of the neighborhood. He derived the Euler–Lagrange equation for the solution Phys. 8.4 Problems 142. A surface in three dimensional space generated by revolving a plane curve about an axis in its plane. Schwarz found the solution of the Plateau problem for a regular quadrilateral in 1865 and for a general quadrilateral in 1867 (allowing the construction of his periodic surface families) using complex methods. Triply Periodic Minimal Surfaces A minimal surface is a surface that is locally area-minimizing, that is, a small piece has the smallest possible area for a surface spanning the boundary of that piece. u a ∇ a ( u b ∇ b η c) + R a b d a b d c u a u d η b = 0, where R a b c d is the Riemann tensor of the ambient space. 2. While these were successfully used by Heinrich Scherk in 1830 to derive his surfaces, they were generally regarded as practically unusable. %PDF-1.5 %���� In the previous step, I have proven that for all h ∈ C 2: ∫ ∫ Δ p ∂ h ∂ x + q ∂ h ∂ y 1 + p 2 + q 2 d x d y = 0. 189 0 obj <> endobj But the integrand F (p) = q 1+|p|2 is not strongly convex, that is D2F δI, only D2F > 0. 8.2 Derivation of MembraneWave Equation 138. By the Young–Laplace equation, the mean curvature of a soap film is proportional to the difference in pressure between the sides. This is equivalent to having zero mean curvature (see definitions below). = 0 Inthiscasewealsosaythat isaminimalsurface. a renewed interest in the theory of minimal surfaces [7]. Oxford Mathematical Monographs. Minimal surfaces can be defined in several equivalent ways in R3. + f 1f 21 f 12+2f 1f 11f 22 = 0 and 1 + f2 2 f 111 2f 1f 11f 11 1 + f2 1 2f 1f 2 f 121 2f 1f hޜѽK�Q��so"d��M�A���m����DS���H��� NJhsP�bK����[-J4�����Z>��s�{Ϲ�c�Ŋ��!Ys�2@*���֠W�S�='}A&�3���+�@�!������2�0�����*��! One might think that if the minimal surface equation had a solution on a smooth domain D ⊂ R n with boundary values φ, it would have a solution with boundary values tφ for all 0 ≤ t ≤ 1. 2 Another revival began in the 1980s. The fact that they are equivalent serves to demonstrate how minimal surface theory lies at the crossroads of several mathematical disciplines, especially differential geometry, calculus of variations, potential theory, complex analysis and mathematical physics.[1]. DIFFERENTIAL EQUATION DEFINITION •A surface M ⊂R3 is minimal if and only if it can be locally expressed as the graph of a solution of •(1+ u x 2) u yy - 2 u x u y u xy + (1+ u y 2) u xx = 0 •Originally found in 1762 by Lagrange •In 1776, Jean Baptiste Meusnier discovered that it … A direct implication of this definition is that every point on the surface is a saddle point with equal and opposite principal curvatures. Seiberg–Witten invariants and surface singularities Némethi, András and Nicolaescu, Liviu I, Geometry & Topology, 2002; What is a surface? The loss of strong convexityor convexity causes non-solvability, or non with the classical derivation of the minimal surface equation as the Euler-Lagrange equation for the area functional, which is a certain PDE condition due to Lagrange circa 1762 de-scribing precisely which functions can have graphs which are minimal surfaces. The criterion for the existence of a minimal surface in $E ^ {3}$ with a given metric is given in the following theorem of Ricci: For a given metric $ds ^ {2}$ to be isometric to the metric of some minimal surface in $E ^ {3}$ it is necessary and sufficient that its curvature $K$ be non-positive and that at the points where $K < 0$ the metric $d \sigma ^ {2} = \sqrt {- K } ds ^ {2}$ be Euclidean. Structures with minimal surfaces can be used as tents. This definition ties minimal surfaces to harmonic functions and potential theory. In this paper, we consider the existence of self-similar solution for a class of zero mean curvature equations including the Born–Infeld equation, the membrane equation and maximal surface equation. Then is a minimal surface if by Example 2.20. Essai d'une nouvelle methode pour determiner les maxima et les minima des formules integrales indefinies. This property is local: there might exist regions in a minimal surface, together with other surfaces of smaller area which have the same boundary. We provide a new and simpler derivation of this estimate and partly develop in the process some new techniques applicable to the study of hypersurfaces in general. uis minimal. One way to think of this "minimal energy; is that to imagine the surface as an elastic rubber membrane: the minimal shape is the one that in which the rubber membrane is the most relaxed. The minimal surface equation is the Euler equation for Plateau's problem in restricted, or nonparametric, form, which can be stated as follows [3, §18.9]: Let fix, y), a single-valued function defined on the boundary C of a simply connected region R in the x — y plane, represent the … derive the minimal surface equation by way of motivation. Catalan proved in 1842/43 that the helicoid is the only ruled minimal surface. Minimal surfaces are part of the generative design toolbox used by modern designers. Soap films are minimal surfaces. Definition 3.2 A smooth surface with vanishing mean curvature is called a minimal surface. ]�[�2UU���%,CR�-qT�4 mY.-����m���Cn�������u���;һm���C�j��+,W��e��{�aO�\C�t�R�Y^�I��\��Fw�+|N�Eaa��|/�����/�6=� �6� [4] Such discretizations are often used to approximate minimal surfaces numerically, even if no closed form expressions are known. This has led to a rich menagerie of surface families and methods of deriving new surfaces from old, for example by adding handles or distorting them. Minimal surfaces necessarily have zero mean curvature, i.e. J. Other important contributions came from Beltrami, Bonnet, Darboux, Lie, Riemann, Serret and Weingarten. Initiated by the work of Uhlenbeck in late 1970s, we study questions about the existence, multiplicity and asymptotic behavior for minimal immersions of closed surface in some hyperbolic three-manifold, with prescribed conformal structure on the surface and second fundamental form of the immersion. Gaspard Monge and Legendre in 1795 derived representation formulas for the solution surfaces. Minimal surface theory originates with Lagrange who in 1762 considered the variational problem of finding the surface z = z(x, y) of least area stretched across a given closed contour. Oxford University Press, Oxford, 2009. xxvi+785 pp. (1 + jr j 2) 1 = = 0: (2) This quasi-linear … The partial differential equation in this definition was originally found in 1762 by Lagrange,[2] and Jean Baptiste Meusnier discovered in 1776 that it implied a vanishing mean curvature.[3]. 9.1 A Difficult Nonlinear Problem 149. Progress had been fairly slow until the middle of the century when the Björling problem was solved using complex methods. Additionally, this makes minimal surfaces into the static solutions of mean curvature flow. 92. The term "minimal surface" is used because these surfaces originally arose as surfaces that minimized total surface area subject to some constraint. An interior gradient bound for classical solutions of the minimal surface equation in n variables was established by Bombieri, De Giorgi, and Miranda in 1968. 0 Derivation of the formula for area of a surface of revolution. Brownian motion on a minimal surface leads to probabilistic proofs of several theorems on minimal surfaces. Abstract. Exercise: (i) Verify the above derivation of the minimal surface equation. By Calabi’s correspondence, this also gives a family of explicit self-similar solutions for the minimal surface equation. Ulrich Dierkes, Stefan Hildebrandt, and Friedrich Sauvigny. A classical result from the calculus of ariations v asserts that if u is a minimiser of A (u) in U g, then it satis es the Euler{Lagrange equation r u. )%-#+'����������������ohdlbjfnaiemckg�����������������8�xeQa����͙=k��ӦN�. Classical examples of minimal surfaces include: Surfaces from the 19th century golden age include: Minimal surfaces can be defined in other manifolds than R3, such as hyperbolic space, higher-dimensional spaces or Riemannian manifolds. h�b"Kv�" ���,�260�X�}_�xևG���J�s�U��a�����������@�������������/ (\$,"*&.! par div. minimal e surfac oblem pr is the problem of minimising A (u) sub ject to a prescrib ed b oundary condition u = g on the @ of. Show that the Euler{Lagrange equation for the ‘surface area’ functional A[v] = Z p 1 + jrvj2 dx (v : !R) is the minimal surface equation div ru p 1 + jruj2 = 0: Problem 3. the Smith conjecture, the Poincaré conjecture, the Thurston Geometrization Conjecture). A direct implication of this definition and the maximum principle for harmonic functions is that there are no compact complete minimal surfaces in R3. ¼ >A7Y>hz á â ã ä Ï B6>AG6\8XY>/W XY:6>)i87958B`AG X \d^ XY:6>m^bZ6G6cAXnstream Mém. This definition makes minimal surfaces a 2-dimensional analogue to geodesics, which are analogously defined as critical points of the length functional. %%EOF Paris, prés. 2. o T do this, e w consider the set U g all tly (su cien smo oth) functions de ned on that are equal to g @. Miscellanea Taurinensia 2, 325(1):173{199, 1760. Jung and Torquato [20] studied Stokes slow through triply porous media, whose interfaces are the triply periodic minimal surfaces, and explored whether the minimal surfaces are optimal for flow characteristics. The complete solution of the Plateau problem by Jesse Douglas and Tibor Radó was a major milestone. ) if and only if f satisﬁes the minimal surface equation in divergence form: div grad(f) p 1 + jgrad(f)j2! In mathematics, a minimal surface is a surface that locally minimizes its area. General relativity and the Einstein equations. [citation needed] The endoplasmic reticulum, an important structure in cell biology, is proposed to be under evolutionary pressure to conform to a nontrivial minimal surface.[6]. Using Monge's notations: p := ∂ f ∂ x; q := ∂ f ∂ y; r := ∂ 2 f ∂ x 2; s := ∂ 2 f ∂ x ∂ y; t := ∂ 2 f ∂ y 2; Where f ∈ C 2 ( Δ ⊂ R 2, R) is the minimal surface (any other function with the same values on the border of Δ has a bigger surface over it). 1.1 Derivation of the Minimal Surface Equation Suppose that ˆRn is a bounded domain (that is, is open and connected). He did not succeed in finding any solution beyond the plane. J. L. Lagrange. 2 f 11f 2! Appendix A: Formulas from Multivariate Calculus 161. Mémoire sur la courbure des surfaces. Lecture 7 Minimal Surface equations non-solvability strongly convex functional further regularity Consider minimal surface equation div √Du 1+|Du|2 = 0 in Ω u = ϕ on ∂Ω. Initiated by … B. Meusnier. Example 3.3 Let be the graph of , a smooth function on . 9.2 Numerical Results 155. This page was last edited on 27 February 2021, at 12:15. Physical models of area-minimizing minimal surfaces can be made by dipping a wire frame into a soap solution, forming a soap film, which is a minimal surface whose boundary is the wire frame. … Thus, we are led to Laplace’s equation divDu= 0. . In architecture there has been much interest in tensile structures, which are closely related to minimal surfaces. One cause was the discovery in 1982 by Celso Costa of a surface that disproved the conjecture that the plane, the catenoid, and the helicoid are the only complete embedded minimal surfaces in R3 of finite topological type. The definition of minimal surfaces can be generalized/extended to cover constant-mean-curvature surfaces: surfaces with a constant mean curvature, which need not equal zero. Minimal surface theory originates with Lagrange who in 1762 considered the variational problem of finding the surface z = z(x, y) of least area stretched across a given closed contour. Acad. In general, This not only stimulated new work on using the old parametric methods, but also demonstrated the importance of computer graphics to visualise the studied surfaces and numerical methods to solve the "period problem" (when using the conjugate surface method to determine surface patches that can be assembled into a larger symmetric surface, certain parameters need to be numerically matched to produce an embedded surface). 9 The KPIWave Equation 149. In the fields of general relativity and Lorentzian geometry, certain extensions and modifications of the notion of minimal surface, known as apparent horizons, are significant. The surface of revolution of least area. This definition uses that the mean curvature is half of the trace of the shape operator, which is linked to the derivatives of the Gauss map. The minimal surface equation is nonlinear, and unfortunately rather hard to analyze. Bernstein's problem and Robert Osserman's work on complete minimal surfaces of finite total curvature were also important. 8 Minimal Surface and MembraneWave Equations 137. Derivation of the Partial Differential Equation Given a parametric surface X(u,v) = hx(u,v),y(u,v),z(u,v)i with parameter domain D, ... For a minimal surface, the eigenvalues of the matrix S are opposites of one another, and thus By viewing a function whose graph was a minimal surface as a minimizing function for a certain area The "first golden age" of minimal surfaces began. Over surface meshes, a sixth-order geometric evolution equation was performed to obtain the minimal surface . Question. the positive mass conjecture, the Penrose conjecture) and three-manifold geometry (e.g. 8.3 Examples 140. The solution is a critical point or the minimizer of inf u| ∂Ω=ϕ Z Ω q 1+|Du|2. 2 the surface M is generated by revolving about the x axis the curve segment y = f(x) joining P 1 - P 2. Example 3.4 The catenoid. 1 in the entire domain, the minimal surface problem is commonly known as Plateau’s Problem [4]. Jn J1 + IY'ul2. Hence the catenoid is a minimal surface. Between 1925 and 1950 minimal surface theory revived, now mainly aimed at nonparametric minimal surfaces. Fix ˚: @!R, and introduce L(;˚) := fu2C0;1(); uj @ = ˚g; (1.1) the set of Lipschitz functions on whose restriction to @ is ˚. Get the full course herehttps://www.udemy.com/course/calculus-of-variations/?referralCode=DCDA4C6662157C098CE5 An equivalent statement is that a surface SˆR3is Minimal if and only if every point p2Shas a neighbourhood with least-area relative to its boundary. Then the Jacobi equation says that. Sci.


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## Due date: October 13

### Problem 1

The IVP

$\displaystyle\begin{cases} x'(t) = \sqrt{x(t)}\\ x(0)=0\end{cases}$

has an infinite number of solutions.

### Problem 2

If

$A=\begin{pmatrix} 1/12 & 5/8\\ 5/8 & 1/12\end{pmatrix}$,

then, for any $x\in\R^2$, $|Ax| \le \dfrac{1}{2}|x|$.

### Problem 3

Let $F(x,t) = \dfrac{tx}{x^2+1}$. Then, for all $t\in\R$,

$|F(x,t) - F(y,t)| \le |t||x-y|$.

### Problem 4

The function $f(x) = \sqrt x$ on $[0,\infty)$ is uniformly continuous but not Lipschitz.

### Problem 5

Consider the operator $\Phi:C([-1,1])\to C([-1,1])$ given by

$\displaystyle \Phi(x)(t) = 1 + 2 \int_0^t s x(s) ds,$

for any $x(t)\in C([-1,1])$. Starting from the constant function $x_0(s)=1$, verify explicitly that the nth iteration of $x_{n+1} = \Phi(x_n)$ is the nth Taylor polynomial of $t\mapsto e^{t^2}$ around $t=0$.


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# Number Theory Problem

Show that $$19^{93}$$ - $$13^{99}$$ is a positive integer divisible by 162.

NOTE:Actually I could not understand how to proceed. Can anyone show me the path ?

Note by Nishant Sharma
4 years, 2 months ago

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What I had done is that I wrote it as $$(18+1)^{93}-(12+1)^{99}$$ and now it is easy to see that a very few terms are left, after the binomial expansion, that are not divisible by $$162$$,i.e., $$3^4*2$$. And then it is very easy to show that the remaining terms add up to be a multiple of $$162$$ as well. · 4 years, 2 months ago

Another method is to note that $$13^2 \equiv 7 \pmod {162}$$. · 4 years, 2 months ago

latex not loading proprly pls type it · 4 years, 2 months ago

A way to reduce the bulk of this problem is to use Euler's totient function. Also, it is inevitably evident that the given number is divisible by $$2$$, so we have to worry about $$81$$ only.

After applying Euler's totient function, we only need to show that: $$19^{39} \equiv c \pmod{81}$$ and $$13^{45} \equiv c \pmod{81}$$ i.e $$19^{39}$$ and $$13^{45}$$ leave the same remainder namely $$"c"$$ on division by $$81$$. · 4 years, 2 months ago

We will have to prove that its divisible by $$81$$ and $$2$$. But its obviously divisible by $$2$$ and if we consider both numbers mod 9, it turns out that its divisible by $$9$$. · 4 years, 2 months ago

Got that it is divisible by 2 and 9. But we need to prove that it is divisible by 162. Anyways, I could not think of approaching it this way. · 4 years, 2 months ago

it goes for 81 also · 4 years, 2 months ago

hey vikram bhaiya please mujhe yei problem latex chor kar likh do mere computer par load nehi ho raha latex · 4 years, 2 months ago

show that 19^93 - 13^99 is a positive integer divisible by 162 · 4 years, 2 months ago

thanks · 4 years, 2 months ago

right but first prove 19^93>13^99 · 4 years, 2 months ago

not needed to show the greater than sign. · 4 years, 2 months ago

yes if we do bionomial then not but otherwise Show that 19^93 - 13^99 is a positive integer · 4 years, 2 months ago

inequality korchhis bole sob jaegae inequality? positivity or negativity is not at all needed as the problem is asking to prove divisibility. · 4 years, 2 months ago

hmm thikie bolecho go sohomda · 4 years, 2 months ago

But why is it required? · 4 years, 2 months ago


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# Relationship between taylor series and geometric series

To find the taylor series of a function you would usually use the formula $\sum_{n=0}^{\infty}\frac{f^{n}(c)}{n!}(z-c)^n$.

However when computing the taylor series for $f(z)=\frac{1}{z+3}$ about $z=1$, I discovered that not only can you compute it using the above formula but you can also compute it using the geometric series formula!

Method 1:

$f(z)=\frac{1}{z+3}, f'(z)=\frac{-1}{(z+3)^2}, f''(z)=\frac{(-1)(-2)}{(z+3)^3}, ..., f^{n}(z)=\frac{(-1)^n n!}{(z+3)^{n+1}}$

$\Rightarrow f^{n}(1)=\frac{(-1)^n n!}{(1+3)^{n+1}} = \frac{(-1)^n n!}{4^{n+1}}$

$\therefore f(z) = \sum_{n=0}^{\infty}\frac{f^{n}(1)}{n!}(z-1)^n = \sum_{n=0}^{\infty}\frac{(-1)^n n!}{4^{n+1}n!} (z-1)^n$

$= \frac{1}{4}\sum_{n=0}^{\infty}\frac{(-1)^n}{4^n} (z-1)^n$ (for $|z-1|<4$)

Method 2:

$f(z)=\frac{1}{z+3}=\frac{1}{(z-1)+1+3}=\frac{1}{(z-1)+4}$

$=\frac{1}{4}\frac{1}{1+\frac{z-1}{4}}$

$=\frac{1}{4}\frac{1}{1-(\frac{-(z-1)}{4})}$

$=\frac{1}{4}\sum_{n=0}^{\infty}(\frac{-(z-1)}{4})^n$ using the formula for geometric series $\frac{1}{1-x}=\sum_{n=0}^{\infty}x^n$, $|x|<1$

$=\frac{1}{4}\sum_{n=0}^{\infty}\frac{(-1)^n}{4^n}(z-1)^n$ (for $|z-1|<4$)

So now my question is, why do they give the same result? Is there some relationship between taylor series and geometric series? Also for what type of functions do they give the same result? (i.e. polynomials, trig functions etc.) Because I found computing the taylor series using the geometric series approach a lot quicker.

The general Taylor series is of the form $$f(x)=\sum_{k=0}^{\infty}{\frac{(x-a)^k}{k!}f^{(k)}(a)}$$
Set $a=0$
$$f(x)=\sum_{k=0}^{\infty}{\frac{x^k}{k!}f^{(k)}(0)}$$
Find some function which satisfies $f^{(k)}(0)=k!$ and you get a geometric series
$$f(x)=\sum_{k=0}^{\infty}{\frac{x^k}{k!}f^{(k)}(0)}=\sum_{k=0}^{\infty}x^k$$
Thus the Taylor series of a function which has infinitely many derivatives at $0$ such that $f^{(k)}(0)=k!$ is a geometric series.


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The following table shows VBA Comparison Operators. Unicode is a For comparing object identities, you can use the keyword is, and its negation is not. Formulas are the key to getting things done in Excel. The general syntax of the not equal to operator is: =IF (cellname <> condition, result 1, result 2) In the if statement, the condition is to check if int_x is not equal to int_y i.e.If int_x is not equal to int_y then if statement should be True, so statement inside the if block should execute, otherwise, else part should:As values of both objects are not equal so condition became True. For example, we might compare the performance of two authors based on a number of articles. To count cells with values greater than, less than or … The not equal operator is a comparison operator in Python. Python not equal is an inbuilt operator returns True if two variables are of the same type and have different values, if the values are identical, then it returns False. Favorite Answer Denominator can be written as (x^2-a^2). We’ll show you a few ways in which you can use the operator with the IF function. Not Equal To. Comparison operators include equal to, less than, greater than and not equal to . We must have used comparison operators in mathematics in the early days. In this accelerated training, you'll learn how to use formulas to manipulate text, work with dates and times, lookup values with VLOOKUP and INDEX & MATCH, count and sum with criteria, … In the table … Ensures that the value of the specified property is not equal to a particular value (or not equal to the value of another property) Example: //Not equal to a particular value RuleFor (customer => customer. In this example, we want to check if intA is not equal to intB. Critical Points Definition of a critical point: a critical point on f(x) occurs at x 0 if and only if either f '(x 0) is zero or the derivative doesn't exist. The result type for these operators is bool. 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Product[Color] <> "Red" Sales[Quantity] <> 0 This means that if one array has a dimension size of zero, then the size of the corresponding dimension in the other array must be 1 or zero, and the size of that dimension in the output is zero. =IF(C2>B2,C2-B2,0) In the above illustration, instead of returning a text result, we are going to return a mathematical calculation. If a string and a number are compared, attempt to convert the string to a number and then check for equality. Excel Formula Training. EG. Python Not Equal Operator To count the number of cells that contain values not equal to a particular value, you can use the COUNTIF function.In the generic form of the formula (above) rng represents a range of cells, and X represents the value you don't want to count.All other values will be counted. You can alternatively just hit the insert Math Region shortcut again (Ctrl + =). Show transcribed image text. We can use set-builder notation: $\{x|x\ge 4\}$, which translates to “all real numbers x such that x is greater than or equal to 4.”Notice that braces are used to indicate a set. This converts whatever is in your equation, back into normal text within your document. I would like to count the number of cells in a column that does not equal zero and then multiply each count by an adjacent number. Null and undefined are equal. The third method is interval notation, in which solution sets are indicated with parentheses or brackets.The solutions to $x\ge 4$ are represented as $\left[4,\infty \right)$. Highlight your does not equal sign, then click the Insert tab and click Equation again. Re: If x does not equal 0 or 1, the expression is equivalent to 30 Jan 2018, 09:06 Display posts from previous: All posts 1 day 7 days 2 weeks 1 month 3 months 6 months 1 year Sort by Author Post time Subject Ascending Descending *If like me, you happen to care not to ignore the existence of matrices without entries (which happens if one or both of their dimensions is$~0$), then this is wrong: the $0\times0$ matrix is both nonsingular (it is the identity) and zero. no of shots days per shot 6 3 0 0 3 5 i want to calculate the "days per shot" column that does not equal zero and then multiply each count by the no. We use these operators to compare different values based on the conditions. The not equal to operator uses the greater than and less than signs together “<>” together. The Not Equal to operator checks if two values are not equal and returns True or False. -10 is the way to write not equal to zero,are you summing or counting? Extrema (Maxima and Minima) Local (Relative) Extrema. We can say that- The total number of articles written by Rajendra > (Greater than)the total number of articles written by Raj. The following are valid expressions using the “not equal to” operator. Question: Prove: If X Is Not Equal To 0 Then (1/x) Is Not Equal To 0 Then X=y This problem has been solved! Here is the example code: Dim intA As Integer Dim intB As Integer Dim blnResult As Boolean intA = 5 intB = 6 If intA <> intB Then blnResult = True Else blnResult = False End If. This is true for all negative values of x therefore not sufficient as x may be -0.5 or -2 Combining two statements x must be negative and still greater than -1 i.e. We can have the following comparison operators in SQL. In other words, restrict it to x ≥ 0 and then we can have an inverse. See the answer. Expert Answer . 1/a^2 b. If either of the operands is NaN, the equality operator returns false. COUNTIF greater than, less than or equal to. a. ! 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## if a is not equal to 0 then a is

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