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# Condition for existence of certain lattice points on polytopes

Let $a_1,\cdots, a_n$ be integers such that $a_i\geq 2$ for all $i$ and $k>0$ another integer.

I am interested in whether there exist integers $x_1,\cdots, x_n$ with $0<x_i<a_i$ satisfying:

$$\frac{x_1}{a_1}+ \cdots + \frac{x_n}{a_n}=k \ \ (*)$$

For example, if $n=2,k=1$, there exist a solution in the specified range iff $\text{gcd}(a_1,a_2) \neq 1$, in other words $\text{lcm}(a_1,a_2)-a_1a_2 \neq 0$.

Question: For what pairs $n,k$ is there a polynomial $F$ whose inputs are $\text{lcm}$ of various subsets of $\{a_i\}_{1\leq i\leq n}$ such that (*) has a solution in the specified range if and only if $F\neq 0$ (see some examples I have in mind below)?

Some observations: If $k\geq n$ then $F\equiv 0$ works, trivially.

Some calculations for small values of $(n,k)$ suggest the following strange function: for each subset $I \subset \{1,\cdots,n\}$ define $$f_I(a_1,\cdots,a_n):= \frac{\prod_{i\in I}a_i}{\text{lcm}\{a_i\}_{i\in I}}$$

For $I=\varnothing$ we set $f_I=1$. Let:

$$F(a_1,\cdots,a_n):= \sum_{I\subset \{1,\cdots,n\}}(-1)^{|I|}f_I(a_1,\cdots,a_n)$$

(You can replace $F$ by a polynomial of the $\text{lcm}$s which vanishes at the same time). This works for $(n,k)=(2,1)$ and probably $(3,1)$ (which would imply $(3,2)$). It does not work for $(4,1)$ but seems to work for some sequence with $(n,k)= (4,2)$ (may be the function depends on $n/k$?). Have anyone seen this kind of formula before in other contexts?

PS: I am not sure what tags should be used. Please feel free to re-tag.

EDIT: Apparently this question is related to existence of lattice points on a polytope. I checked through some of the references pointed to in this question on MO, but could not find the exact answer to what I wanted.

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Could you elaborate on "probably (3,1)?" – Douglas Zare Jan 28 '10 at 10:16
@Douglas: I used brute force but did not check everything carefully. – Hailong Dao Jan 28 '10 at 11:21
I think I'll delete my "answer" below, since it doesn't seem to be of value to this problem. Although, I'll leave the remark that if the $x_i$ in (*) are allowed to be chosen from the set of integers. Then you can choose $x_2=x_3=\cdots=x_n=0$ and $x_1=ka_1$ to get a solution. – Douglas S. Stones Jan 31 '10 at 5:29

 I don't know if you're still interested in this problem Hailong, but here is a partial result. I make two natural restrictions :

• Restriction 1 : avoid small primes. Let $B_n$ denote the (infinite) set of integers all of whose prime factors are $>n$. I assume that the all the $a_i$ are in $B_n$ (this is to avoid the difficulty that for example when $n=3$ and $k=1$, (*) has a solution when $a_1=a_2=a_3=p$ for $p$ a prime $>2$ , but not when $a_1=a_2=a_3=2$ ).

• Restriction 2 : avoid small $k$. I assume that $k \geq \frac{n}{2}$ (this is to avoid the difficulty that for example when $n=4$ and $a_1=a_2=2,a_3=a_4=3$, (*) has a solution for $k=2$ but not for $k=1$). Under those restrictions, the following conditions are equivalent :

(i) (*) has a solution in the desired range. (ii) No $a_i$ is prime to all the others $a_j$. (iii) The polynomial $F=\prod_{i=1}^{n}G_i$ is nonzero, where $G_i$ is the polynomial $\sum_{j\neq i}(a_ia_j-\text{lcm}(a_i,a_j))$.

Note that the polynomial is independent of $k$.

The only difficult implication is $(ii) \rightarrow (i)$. To show this, consider the undirected graph $G$ whose vertices are the integers from $1$ to $n$ and such that there is an edge joining $i$ to $j$ iff $gcd(a_i,a_j)>1$. Then condition (ii) says that $G$ is connected. By a straightforward graph-theoretic lemma, there is a subgraph of $G$ which is a disjoint union of stars. Thus, we can write $\lbrace 1,2, \ldots n\rbrace$ as a disjoint union $A_1 \cup A_2 \cup \ldots \cup A_t (t \geq 1)$ such that for each $l$ between $1$ and $t$ we have $|A_l| \geq 2$ and there is a distinguished vertex $u_l$ in $A_l$ that is connected to all the other vertices in $A_l$. Restriction 2 ensures that we can find a decomposition $k=\sum_{l=1}^{t}\alpha_l$ where each $\alpha_l$ is an integer with $0<\alpha_l < |A_l|$. Restriction 1 ensures that we may find, for each $l$ $(x_i)_{i\in A_l}$ such that $0<x_i<a_i (i\in A_l)$ and $\sum_{i\in A_l}\frac{x_i}{a_i}=\alpha_l$. So we are done.

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Dear Ewan, I am still very interested! I kind of need a formula to work without extra conditions, but your result is very nice. Thanks and +1. – Hailong Dao Feb 11 '10 at 1:16
Actually, I am now confused, (iii) seems to be equivalent to $gcd(a_i,a_j)>1$ for all pair $i\neq j$? – Hailong Dao Feb 11 '10 at 6:31
You're right, I interchanged $+$ and $\times$ : $F$ is the product of the $G_i$'s, not the sum, and $G_i$ is a sum not a product. I'll correct this in my answer. The polynomial $G_i$ is always positive, and it is equal to 0 iff all the $a_ia_j-\text{lcm}(a_i,a_j)$ are zero, iff $a_i$ is prime to all the other $a_j$'s. – Ewan Delanoy Feb 11 '10 at 9:24


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# PLATE-Seq for genome-wide regulatory network analysis of high-throughput screens

## Abstract

Pharmacological and functional genomic screens play an essential role in the discovery and characterization of therapeutic targets and associated pharmacological inhibitors. Although these screens affect thousands of gene products, the typical readout is based on low complexity rather than genome-wide assays. To address this limitation, we introduce pooled library amplification for transcriptome expression (PLATE-Seq), a low-cost, genome-wide mRNA profiling methodology specifically designed to complement high-throughput screening assays. Introduction of sample-specific barcodes during reverse transcription supports pooled library construction and low-depth sequencing that is 10- to 20-fold less expensive than conventional RNA-Seq. The use of network-based algorithms to infer protein activity from PLATE-Seq data results in comparable reproducibility to 30 M read sequencing. Indeed, PLATE-Seq reproducibility compares favorably to other large-scale perturbational profiling studies such as the connectivity map and library of integrated network-based cellular signatures.

## Introduction

High-throughput screening (HTS) represents a key component of drug discovery and a critical technology used throughout biomedical research1, 2. Due to cost and complexity, however, most screens are still performed using low-complexity reporters, such as cell viability, inhibition of specific enzymes, binding affinity, protein-protein interactions, mitochondrial respiration, and cell growth. In contrast, there is increasing interest in coupling HTS with genome-wide reporters, which may provide a more comprehensive portrait of drug activity2,3,4.

Indeed, a key advantage of genome-wide reporters is their more universal nature; specifically, the ability to test hypotheses that may not have been considered at the time of assay development. For instance, the same data from gene expression profiling of cellular perturbations have been used effectively to predict compound mechanism-of-action (MoA)5,6,7 and sensitivity in specific cellular contexts8, as well as to identify synergistic drug combinations9, 10, compounds with similar MoA11, and candidates for drug repositioning7. Indeed, when incorporated in an HTS setting, genome-wide profiles can report on virtually any genes or pathways of interest, without requiring an a priori commitment.

Unfortunately, due to their relatively high cost and labor-intensive nature, genome-wide expression profiles have not been incorporated as primary reporters in HTS campaigns. A few exceptions, such as connectivity map (CMap)2, represent proof-of-concept studies rather than scalable approaches and have either been restricted to a handful of cell lines or replaced by methodologies that report on a limited number of genes (e.g., Luminex L1000 reporters)3.

To address this challenge, we introduce a new approach that combines a highly scalable and multiplexed RNA-Seq protocol (PLATE-Seq) with regulatory network analysis. Collectively, this integrative and fully automated pipeline allows accurate, reproducible characterization of the proteins, whose activity is affected by a library of bioactive compounds. The proposed approach involves two key concepts: (a) a strategy for barcoding and pooling cDNA libraries to substantially reduce the cost and complexity of multi-sample RNA-Seq and (b) the use of network-based algorithms for the highly reproducible inference of protein activity from low-depth RNA-Seq profiles (0.5–2 M reads). Taken together, this combination supports a >tenfold cost reduction with virtually no reduction in assay accuracy and reproducibility, compared to standard depth (30 M read) sequencing. This translates into a dramatic increase in gene reporter dimensionality for HTS applications—from a few observables to a genome-wide repertoire—at a total reagent cost of ~$15 per sample. ## Results ### PLATE-Seq technology Recent advances in multiplexed and single-cell RNA-Seq have led to significant increases in cost effectiveness and scalability of gene expression profiling12,13,14,15,16,17. These methodologies introduce sample-specific sequence barcodes into cDNA prior to library construction, allowing early pooling of cDNA from many samples and a proportional decrease in reagent and labor costs. They are also optimally suited to automation, allowing straightforward integration into HTS pipelines for analysis of RNAi or small molecule perturbations in a multi-well format. Here we use automated liquid handling to introduce lysis buffer, capture polyadenylated mRNA with an oligo(dT)-grafted plate, and deliver well-specific, barcoded oligo(dT) primers to every sample in a multi-well plate (Fig. 1a). After reverse transcription, the cDNA in each well contains a specific barcode sequence on its 5′ end and a common adapter, such that all samples can be combined into a single pool for purification and concentration. We then use Klenow large fragment for pooled second-strand synthesis from adapter-linked random primers. Because this polymerase lacks strand displacement and 5′ to 3′ exonuclease activities, each cDNA molecule produces at most, one second-strand synthesis product containing the sample barcode (Fig. 1b). Finally, the pooled library is enriched in a single PCR prior to sequencing. The resulting libraries represent the 3′ ends of mRNAs and are sequenced to a depth of 0.5–2 M raw reads per sample. ### PLATE-Seq performance To characterize the performance of PLATE-Seq, we conducted a fully automated, 96-well screen to profile BT20 breast cancer cells following treatment with seven well-characterized small-molecule perturbagens (plus DMSO controls) and 12 replicates per condition. Figure 2a, b shows the detected gene and uniquely mapped read distributions across the 96-well plate samples, respectively. On average, we detected ~10,200 genes per sample from ~670,000 uniquely mapped reads from ~10,000 cells per well. To compare PLATE-Seq to conventional RNA-Seq, we processed some of the replicates using the Illumina TruSeq protocol, to a standard depth of ~30 M reads per sample. Critically, TruSeq required pooling six samples to meet the minimum cDNA input requirements. Figure 2c, d shows the detected gene and uniquely mapped read distributions for both TruSeq and PLATE-Seq samples, normalized to the same number of wells by pooling data from the six PLATE-Seq replicates. Figure 2e, f shows the saturation behaviors of individual PLATE-Seq wells, TruSeq samples, and aggregate PLATE-Seq data pools equivalent to the TruSeq replicate pools. Analysis of data pooled from six single-well PLATE-Seq profiles detected >75% of the genes that had been detected in the same number of replicates by TruSeq, albeit with sixfold fewer reads. We also compared the relative efficiencies of conventional column isolation and oligo(dT) bead-based purification of mRNA and the oligo(dT) plate-based purification used here. Not only does the plate-based purification offer significant advantages in terms of automation, the gene detection efficiencies and saturation behaviors of the resulting libraries from the two methods are essentially identical (Supplementary Fig. 3). We also investigated whether the gene expression signatures associated with each drug were comparable between the two methods. We used DESeq2 (ref. 18) to identify differentially expressed genes (q < 0.05) between each set of drug-treated and vehicle-control replicates measured by PLATE-Seq and projected a matrix of fold changes computed from both PLATE-Seq and TruSeq across these genes using multidimensional scaling (Fig. 2g). As expected, replicates associated with distinct drug treatments cluster together, consistent with their tissue-specific activity and independent of profiling technology. For instance, drugs such as aprepitant, gemcitabine, and albendazole, which had minimal effects on BT20 cells, were clustered together. In contrast, strongly bioactive drugs, such as crizotinib and bortezomib, produced distinct but highly reproducible and technology-independent clusters. Drugs with similar MoA, such as the topoisomerase II inhibitors mitoxantrone and idarubicin, clustered together. Finally, we performed gene set enrichment analysis (GSEA) of significantly altered genes detected by PLATE-Seq in the TruSeq differential expression analysis. For all gene sets, we found highly significant enrichment of the PLATE-Seq genes in the appropriate direction (Supplementary Fig. 1). Hence, both PLATE-Seq and conventional RNA-Seq produce comparable gene expression signatures following drug perturbation. In Alvarez et al.19, we showed that assessing protein activity from gene expression profiles using VIPER is highly robust to low-depth sequencing. Indeed, unlike gene expression profiles, whose correlation degraded rapidly with decreasing sequencing depth, Spearman’s correlation of VIPER-inferred activity profiles was virtually unaffected between 0.1 and 30 M reads19. This is because VIPER infers protein activity from expression of its tissue-specific transcriptional targets, as identified by the ARACNe algorithm. Thus, VIPER integrates dozens to hundreds of gene expression measurements to infer activity of a single protein, effectively averaging out noise resulting from low-depth sequencing19. ### Protein activity inference with PLATE-Seq To demonstrate the practical utility of this theoretical advantage when using low-depth PLATE-Seq assays, we used VIPER to compute differential activity of ~6000 regulatory proteins, including ~2000 transcription factors and ~4000 signaling proteins, following treatment with the topoisomerase II inhibitors mitoxantrone and idarubicin19. Figure 2h shows the gene expression signatures of the top 40 genes most differentially expressed following perturbation by either drug, based on PLATE-Seq. As expected from Fig. 2g, there is excellent agreement between PLATE-Seq and TruSeq. To identify key regulatory proteins responsible for the pharmacological effects of these two drugs and thus comprising their MoAs, we analyzed both the PLATE-Seq and the TruSeq profiles with VIPER. We used our recently published breast cancer-specific gene regulatory network19 generated by ARACNe analysis20, 21 of gene expression profiles from the breast cancer cohort collected by The Cancer Genome Atlas consortium22. Figure 2i shows the top 40 most differentially activated proteins following mitoxantrone or idarubicin treatment, based on PLATE-Seq. Critically, VIPER-inferred protein activity was highly reproducible across PLATE-Seq and TruSeq samples (see Supplementary Fig. 2 for GSEA). Furthermore, TOP2A, the gene that encodes the target of both drugs, was among the most deactivated regulatory proteins by VIPER analysis, along with several other key cell cycle control proteins (e.g., CEPNF, FOXM1, CCNA2, BUB1B, and CENPK). Finally, Fig. 2j shows the gene expression signatures associated with these top 40 differentially active proteins. Consistent with previous studies19, 23, 24, many of these proteins, including cell cycle regulators, that play an important role in the MoA of these drugs, such as FOXM1, CCNA2, and TOP2A, are not among the top 40 most differentially expressed genes. This effectively demonstrates that the combination of PLATE-Seq and VIPER analysis produces differential protein activity profiles that are remarkably robust, independent of the low-sequencing depth. ### Comparison of PLATE-Seq to CMap and LINCS Previous efforts have attempted to elucidate MoA based on gene expression alterations in cell lines2, 7. CMap represents a first seminal attempt at this approach based on microarray profiling of cell lines treated with several hundred compounds2. To increase scalability, library of integrated network-based cellular signatures (LINCS) implemented the Luminex L1000, which reports on ~1000 representative markers from which broader expression profiles could be computationally inferred3, 25. However, these two data sets, which represent the current state-of-the-art, require a trade-off between cost/efficiency and coverage. Therefore, we reasoned that PLATE-Seq might provide a solution to this trade-off by supporting a genome-wide reporter assay while leveraging the broader dynamic range of RNA-Seq compared to microarrays and the ability to profile >tenfold more genes than the L1000 at a comparable cost. To compare PLATE-Seq to these technologies, we conducted a 4 × 96-well screen with 184 clinically relevant compounds plus vehicle controls (Supplementary Table 1) in U87 cells in duplicate. We then systematically compared the gene-level noise distribution we obtained to those of the CMap and LINCS/L1000 data sets. Figure 3a–c shows how the average coefficients of variation across replicates vary with average expression level for PLATE-Seq in comparison to CMap data obtained with the GPL3921 microarray platform, CMap with the GPL96 platform, and LINCS/L1000, respectively. Similarly, Fig. 3d–f shows the overall distributions of average coefficients of variation for these same comparisons. Despite the significantly lower costs associated with PLATE-Seq, we obtain comparable noise distributions to CMap. In addition, we point out that although the noise in the LINCS data set is twofold lower than in the PLATE-Seq screen (for duplicates taken from GSE70138), measurements are only made for 978 genes (>tenfold fewer than in PLATE-Seq, which is a genome-wide measurement). We also show that the dynamic range for signal in PLATE-Seq is comparable to these alternative methods by calculating the fold-change distribution across each expression measurement and perturbation relative to vehicle controls (Supplementary Fig. 4). Hence, PLATE-Seq compares favorably to previous approaches and may offer an effective, low-cost platform for assessing compound MoA similarity. ## Discussion Small molecule perturbations induce complex, cell-context-specific alterations that ultimately affect thousands of genes. This challenges the use of low-complexity reporter assays in HTS to elucidate MoA. In contrast, RNA-Seq represents a high-dimensional assay that is particularly well suited for compound MoA elucidation19, 26, similarity analysis2, 7, and synergy analysis9 especially when using network-based methods that may pinpoint the key regulatory targets and effectors. However, conventional RNA-Seq involves independent library construction for each sample and deep sequencing, making its application to HTS prohibitively expensive and poorly suited to automation. For instance, at an average cost of$300 per profile, screening a library of ~4000 compounds across two time points, concentrations, and replicates would cost ~$10 M in reagents. However, when combined with automated, pooled library preparation and the tenfold reduction in sequencing depth made possible by leveraging VIPER, the reagent cost for the same 4000-compound campaign would be less than$0.5 M, a 20-fold reduction.

A possible limitation of the approach is its dependence on regulon quality for accurate inference of protein activity. Using large-scale benchmarks, we have shown that differential activity is accurately predicted for >70% of transcriptional regulators and ~60% of signaling proteins19. However, if specific proteins of interest are poorly characterized in the assay, ad hoc regulons can be generated by combining DNA-binding assays and expression profiles following perturbation27. In addition, as computational reverse engineering methods improve, accuracy of protein activity inference will increase. In this case, data from previous screens can simply be re-analyzed using the improved models to produce more accurate and extensive protein-activity profile coverage, thus avoiding obsolescence of PLATE-Seq data sets.

We expect PLATE-Seq to have broad pharmacological applications, including elucidation of compound MoA and synergy, support for drug re-positioning, and precision medicine approaches where MoA is matched to patient-specific disease dependencies. Finally, while we focused mainly on drug screening, PLATE-Seq is equally amenable to RNAi, CRISPR-Cas9, and cDNA screening assays. This highly scalable and economical approach integrates HTS with high-dimensional analysis of RNA expression and protein activity, revealing the connections between cellular perturbations and genome-wide regulatory interactions.

## Methods

### Cell culture and drug perturbation assays

BT20 epithelial breast carcinoma cells were cultured in white 96-well tissue culture-treated plates (Greiner 655083) at a starting density of 8000 cells per well in 100 μl of eagle’s minimum essential medium supplemented with 10% fetal bovine serum (FBS) and 1% penicillin/streptomycin. After 24 h of incubation, the plates were treated with drugs. Each drug was dosed at the concentration at which the cells were 80% viable after 48 h of treatment. After 6 h of treatment, the medium was replaced with 100 ml of FBS supplemented with 10% DMSO and the plates were frozen at −80 °C prior to PLATE-Seq. After thawing for PLATE-Seq, cells were washed twice with phosphate-buffered saline (PBS) prior to lysis.

### Automated PLATE-Seq

Cells are lysed for PLATE-Seq in Buffer TCL (Qiagen) supplemented with 2-mercaptoethanol in 96-well plates. We transfer 30 μl of cell lysate from each well to a 96-well plate with oligo(dT) grafted to the wells (mRNA TurboCapture Plate, Qiagen) and added 1 μl of 1:100 dilution of ERCC Ex-Fold Spike-Ins (ThermoFisher) to a subset of the sample wells. The plate is then centrifuged at 295 × g for 1 min to eliminate air bubbles followed by incubation for 90 min at room temperature with shaking at 200 rpm. Lystate is then removed from the oligo(dT) capture plate and wells are washed three times with Buffer TCW (Qiagen) using the 96 CO-RE Head of a Hamilton Microlab STAR automated liquid handling system. After drying the plate, mRNA is then eluted by adding 30 μl of elution buffer (nuclease-free water supplemented with 0.4 U μl−1 SUPERaseIN (ThermoFisher)), incubation at 65 °C for 5 min, and incubation at room temperature for 5 min. mRNA samples are then transferred to a new PCR plate (Hard-Shell PCR plate, 96-well, Bio-Rad) using sterile-filter 96-well, 50 μl tips on the Hamilton STAR.

To anneal primers for reverse transcription, 3 μl of 100 μM adapter-linked oligo(dT) primers are added to each well along with 10 μl of 5× ProtoScript RT Buffer (New England BioLabs) with the Hamilton STAR CO-RE Head and 1000 μl channel pipetting system (see Supplementary Table 2 for “PLATEseq_oligodTBC” sequences). The plate is then heated to 94 °C for 2 min and placed immediately on ice for 5 min. The remaining components of the reverse transcription master mix including 0.5 mM of each dNTP, 10 mM DTT, 35.8 U of Protoscript II Reverse Transcriptase (New England BioLabs), and water are then added to bring the final reaction volume to 50 μl per well. The plate is then centrifuged at 295 × g for 1 min and incubated at 42 °C for 2 h. At this point, the samples can optionally be frozen at −80 °C.

To remove excess primer, 1 μl of a fourfold dilution of Exonuclease I (New England BioLabs) is added to each sample. The Exonculease I reaction is incubated at 25 °C for 1 h. To hydrolyze the RNA, we add 20 μl of a 1:1 mixture of 1 M sodium hydroxide and 0.5 M EDTA to each well and incubate at 65 °C for 15 min. The wells are then pooled together into a single tube to which 80 μl of 12 M hydrochloric acid are added to neutralize the sample. At this point, the pooled sample can optionally be frozen at −80 °C.

The pooled, neutralized sample is purified and concentrated using a Qiagen MinElute column using a vacuum apparatus according to the manufacturer’s instructions, and pooled cDNA is eluted in 15 μl of nuclease-free water. For second-strand synthesis, 1 μl of 10 mM dNTP mixture and 1 μl of 100 mM adapter-linked random hexamer primers are added to the pooled cDNA sample (see Supplementary Table 2 for “PLATEseq_second_strand” primer sequence). The mixture is then heated to 70 °C for 2 min and immediately placed on ice for 5 min. Next, 2 μl of NEB Buffer 2 (New England BioLabs) and 1 μl of Klenow large fragment DNA polymerase (New England BioLabs) are added and the reaction is incubated at room temperature for 30 min. The reaction is then stopped by addition of EDTA to a final concentration of 50 μM the double-stranded cDNA is purified with two rounds of bead clean-up using AMPure XP beads (Beckman Counter) at a 1:1 bead-to-sample ratio. Finally, the double-stranded cDNA pool is amplified by PCR using Phusion DNA Polymerase (New England BioLabs) with 0.5 μM Illumina RP1 primer and Illumina RPIx primer (where x is a number indicating the Illumina index). The thermocycling protocol begins with 98 °C for 30 s followed by 10–12 cycles of 98 °C for 15 s, 62 °C for 15 s, and 72 °C for 60 s followed by a 7-min incubation at 72 °C.

Samples are sequenced on an Illumina NextSeq 500 sequencer using 75 cycle version 2 sequencing kits. Custom sequencing primers were used at a final concentration of 300 nM. Read 1 comprises 26 cycles and read 2 comprises either 60 or 66 cycles depending upon whether or not an index read is included. In each experiment, 20–30% PhiX library is added to the sample, which is clustered at a final concentration of 1.6 pM.

### Comparison of PLATE-Seq to conventional RNA-Seq

To compare PLATE-Seq and conventional RNA-Seq, we conducted a small-scale screen in a single 96-well plate. We treated each well containing cultured BT20 cells with one of seven compounds or DMSO (vehicle), including mitoxantrone, crizotinib, albendazole, idarubicin, aprepitant, gemcitabine, and bortezomib, with 12 replicate wells for each condition. Half of the material in each well was allocated to PLATE-Seq, and 96 libraries were constructed using the automated, pooled procedure described above. For the other half, we pooled two sets of six wells for each condition to generate 16 samples in total (two replicates for each condition). We then extracted total RNA from each pool using an RNeasy column (Qiagen) and subjected the total RNA to Illumina TruSeq poly(A) + RNA-Seq library construction according to the manufacturer’s instructions. The resulting 16 RNA-Seq libraries were then sequenced to a depth of ~30 M raw, 100-base single-end reads on an Illumina HiSeq 2500 sequencer.

### PLATE-Seq performance with conventional mRNA isolation

LnCAP human prostate adenocarcinoma cells were plated on a 96-well tissue culture treated plate (Greiner #655083) at a starting density of 10,000 cells per well in 100 μl of medium (RPMI1640 supplemented with 10% FBS). After 24 h of incubation, the medium was removed and cells were lysed in 30 μl of Buffer TCL (Qiagen) with 1% β-mercaptoethanol. A measure of 15 μl of this lysate was transferred to a new plate and used for automated PLATE-Seq; the remaining 15 μl of lysate in each well was mixed with 135 μl of Buffer RLT (Qiagen). A measure of 150 μl of 70% ethanol was added to each well in this plate and all 300 μl was transferred onto the RNeasy 96 filter plate (Qiagen) for total RNA extraction according to the manufacturer’s instructions, including DNase digestion. The first elution from the columns was in 50 μl of RNase-free water, and the second elution was in 45 μl of RNase-free water.

RNA Purification Beads (Illumina) were used to isolate poly(A)-containing mRNA molecules. Beads were warmed to room temperature and washed three times with lysis/binding buffer (20 mM Tris-HCl, pH 7.5, 500 mM LiCl, 0.5% LiDS, 1 mM EDTA, 5 mM DTT) and re-suspended in the original starting volume with lysis/binding buffer. We then added 50 μl of washed beads to each well of eluted RNA from above and followed the manufacturer’s protocol (Illumina TruSeq Stranded mRNA Sample Preparation Guide). Instead of using Illumina’s Fragment, Prime, Finish Mix, we eluted mRNA from the beads with 16.5 μl of Elution solution (20 mM Tris-HCl pH 7.5, 1 mM EDTA). A measure of 1 μl of 1:5000 diluted ERCC Ex-Fold Spike-Ins (ThermoFisher) was added to half of the sample wells. We used this as input for the automated PLATE-Seq protocol, beginning with the addition of 5× Protoscript RT Buffer and 1.5 μl of 100 μM adapter-linked, barcoded oligo(dT) primers to each well.

### PLATE-Seq data processing

PLATE-Seq data comprise sets of two paired-end reads. The first read contains an eight-base barcode sequence that identifies the well from which a given sample originates. The second read contains a sequence that typically maps strand specifically to the 3′ end of an mRNA transcript. We first record the barcode sequence associated with each read. The barcode sequences are designed so that a single error can be corrected, and so we allow an edit distance of one from the actual barcode sequences. Next, we use the bwa-mem aligner to map the set of second reads to a pre-assembled human transcriptome (hg19, UCSC known genes). We then demultiplex the resulting alignments and count the number of uniquely mapped reads associated with each gene. We define a read as uniquely mapped if the maximum alignment score among alignments that map to the correct strand is associated with a single gene.

### Gene expression analysis from PLATE-Seq

The differential expression analysis shown in Fig. 2i was conducted using DESeq2 using a negative binomial model18. Differentially expressed genes (q < 0.05) from all comparisons of drug-treated vs. vehicle control samples were identified exclusively from the PLATE-Seq data. A log-scale fold-change matrix was then generated for this set of differentially expressed genes across all PLATE-Seq and TruSeq samples and clustered using multidimensional scaling (MDS) with Euclidean distance as implemented in MATLAB.

For all other analyses, gene expression signatures for drug perturbations were obtained by first variance stabilizing the raw read counts with DESeq2 (ref. 18) and then subtracting the mean vehicle control expression level from the perturbation expression levels.

### Regulatory network analysis from PLATE-Seq

Gene expression signatures were computed for each individual sample by first variance stabilizing the gene expression data with the DESeq package, and then subtracting the average of the vehicle control samples from each drug treatment profile. Relative protein activity was inferred from the single-sample gene expression signatures using the VIPER algorithm19.

### Large-scale PLATE-Seq screen in U87 cells

U87 cells were plated in a 384-well plate at 200,000 cells ml−1 in 20 μl of media for a final count of 4000 cells per well. After 24 h, the Labcyte Echo 550 Liquid Handling system was used to dispense drugs at concentrations resulting in 20% growth inhibition (Supplementary Table 1). After 24 h of drug treatment, media was removed and cells were washed with PBS. Cells were then lysed in 20 μl of Buffer TCL (Qiagen) containing 1% β-mercaptoethanol. Following the z-pattern of quadrants for a 384-well plate, 20 μl of lysate from quadrant 1, and 20 μl of lysate from quadrant 3 were pooled into one well of a 96-well TurboCapture plate (mRNA TurboCapture Plate, Qiagen) for a total of 40 μl of lysate in each well during the mRNA capture step. Automated PLATE-Seq was then followed as above. The resulting pooled libraries were sequenced on an Illumina HiSeq 4000 using a 50-cycle SBS kit and a PE cluster kit. Read 1 comprised 18 cycles and Read 2 comprised 50 cycles.

### Comparison of PLATE-Seq screens to CMap and LINCS

In Fig. 3, we compare the coefficient of variation (CV) between PLATE-Seq, CMap, and LINCS screening data sets. For PLATE-Seq, we took our U87 screen (4 × 96 wells with 184 drugs plus controls) data set and computed the CV and average expression for each gene in each duplicate measurement. We then plotted the average CV vs. average expression across all duplicates for each gene in Fig. 3a–c and distribution of these same average CVs in Fig. 3d–f. For CMap, we took all duplicates from screens conducted in MCF7 (processed average difference values available on the Gene Expression Omnibus under accession GSE5258) and applied the same calculations as we did for PLATE-Seq. We analyzed microarray data generated using the GPL3921 and GPL96 platforms separately. Finally, we obtained the LINCS data set from the Gene Expression Omnibus under accession GSE70138 (Level 2 data corresponding to gene expression values for genes that were measured directly) for ~26,000 randomly selected duplicates and applied the same calculation to the ~1000 genes measured directly on the L1000 platform. In Supplementary Fig. 4, we compare PLATE-Seq, CMap, and LINCS in terms of the fold change in each gene expression measurement across all perturbations relative to their respective vehicle controls. We calculated these distributions using the same expression matrices that we used for the CV comparisons.

### Note on cells lines

We note that BT20 cells are on the ICLAC list of commonly misidentified cell lines, but that the identity of a particular cell line is immaterial to the conclusions of our technical assessment of PLATE-Seq. Nonetheless, the BT20, U87, and LnCAP cell lines used these studies were all obtained from ATCC, which authenticates the cell lines it provides by STR testing. Internally, we tested all of the cell lines for mycoplasma prior to use.

### Code availability

The VIPER algorithm is available at http://califano.c2b2.columbia.edu/viper/.

### Data availability

All of the raw and processed data associated with this study are available at the Gene Expression Omnibus under accession GSE97460.

## References

1. Macarron, R. et al. Impact of high-throughput screening in biomedical research. Nat. Rev. Drug. Discov. 10, 188–195 (2011).

2. Lamb, J. et al. The connectivity map: using gene-expression signatures to connect small molecules, genes, and disease. Science 313, 1929–1935 (2006).

3. Vempati, U. D. et al. Metadata standard and data exchange specifications to describe, model, and integrate complex and diverse high-throughput screening data from the library of integrated network-based cellular signatures (LINCS). J. Biomol. Screen. 19, 803–816 (2014).

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9. Bansal, M. et al. A community computational challenge to predict the activity of pairs of compounds. Nat. Biotechnol. 32, 1213–1222 (2014).

10. Mitrofanova, A. et al. Predicting drug response in human prostate cancer from preclinical analysis of in vivo mouse models. Cell Rep. 12, 2060–2071 (2015).

11. Wolpaw, A. J. et al. Modulatory profiling identifies mechanisms of small molecule-induced cell death. Proc. Natl Acad. Sci. USA 108, E771–E780 (2011).

12. Hashimshony, T., Wagner, F., Sher, N. & Yanai, I. CEL-Seq: single-cell RNA-Seq by multiplexed linear amplification. Cell Rep. 2, 666–673 (2012).

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15. Macosko, E. Z. et al. Highly parallel genome-wide expression profiling of individual cells using nanoliter droplets. Cell 161, 1202–1214 (2015).

16. Klein, A. M. et al. Droplet barcoding for single-cell transcriptomics applied to embryonic stem cells. Cell 161, 1187–1201 (2015).

17. Bose, S. et al. Scalable microfluidics for single-cell RNA printing and sequencing. Genome Biol. 16, 120 (2015).

18. Love, M. I., Huber, W. & Anders, S. Moderated estimation of fold change and dispersion for RNA-seq data with DESeq2. Genome Biol. 15, 550 (2014).

19. Alvarez, M. J. et al. Functional characterization of somatic mutations in cancer using network-based inference of protein activity. Nat. Genet. 48, 838–847 (2016).

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21. Margolin, A. A. et al. ARACNE: an algorithm for the reconstruction of gene regulatory networks in a mammalian cellular context. BMC Bioinformatics 7, S7 (2006).

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## Acknowledgements

P.A.S. was supported by Grant K01EB016071 from NIH/NIBIB. A.C. was supported by grant R35CA197745 from NIH/NCI. A.C. was also supported by grants S10OD021764 and S10OD012351 from the Office of the Director of NIH, and both A.C. and P.A.S. were supported by grant U54CA209997 from NIH/NCI.

## Author information

Authors

### Contributions

PLATE-Seq was conceived and designed by A.C. and P.A.S. Experimental methodology for PLATE-Seq was developed by E.C.B., F.R. and P.A.S. Analytical methodology for network-based analysis of PLATE-Seq was developed by M.J.A. and A.C. Data processing and analysis were conducted by E.C.B., F.R., M.J.A., P.A.S. and A.C. HTS experiments were conducted by F.R., R.R., H.L. and C.K. Automated PLATE-Seq was developed and implemented by R.R., C.K. and E.C.B. E.C.B., F.R., M.J.A., R.R., H.L., C.K., A.C. and P.A.S. wrote the paper.

### Corresponding authors

Correspondence to Andrea Califano or Peter A. Sims.

## Ethics declarations

### Competing interests

M.J.A. is chief scientific officer of DarwinHealth Inc. A.C. is a founder of DarwinHealth Inc. The remaining authors declare no competing financial 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

Bush, E.C., Ray, F., Alvarez, M.J. et al. PLATE-Seq for genome-wide regulatory network analysis of high-throughput screens. Nat Commun 8, 105 (2017). https://doi.org/10.1038/s41467-017-00136-z

• Revised:

• Accepted:

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• DOI: https://doi.org/10.1038/s41467-017-00136-z

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• ### 3’Pool-seq: an optimized cost-efficient and scalable method of whole-transcriptome gene expression profiling

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# A clock travelling “faster than the speed of light” [closed]

When a clock travels close to the speed of light as observed by us, it should tick slower.

Does this mean that, when using this clock to measure a second - it could travel much more than 300 000 km during a second? In fact, this clock could travel infinitely fast when viewed only from its own perspective?

Does this imply that if I traveled on a space ship, at the speed of 99 % of light, I would age 4 years, but could reach a destination perhaps 400 light years away?

My friends and family on earth would have perished, but the people I saw on the destination planet as I started my journey will have aged only 4 years as well.

If the last assumption is correct; it means that we could watch a launching space ship at alpha centauri, it could arrive here in just 3-4 days - appearing to travel much faster than the speed of light, without breaking the laws of relativity. We would just say that it was relativistic blue shift that made it seem to travel faster than the speed of light.

-

## closed as off-topic by Gerald, Eduardo Serra, frodeborli, Manishearth, TildalWaveFeb 15 at 10:49

• This question does not appear to be about astronomy, within the scope defined in the help center.
If this question can be reworded to fit the rules in the help center, please edit the question.

Yes, you would age by four years and travel very very far and everyone you know would get 400000 years older. However, you, having stopped at that distant location would have to wait for about 400000 years for these events to unfold as you point your powerful telescope at the Earth. Remember that when you look far you see the past, but not the present. –  Alexey Bobrick Feb 10 at 15:32
This seems more like a general physics question than an astronomy question. Sure, it uses a space journey as an example (which actually sounds more like Space Exploration) but the heart of the question is general physics. –  called2voyage Feb 10 at 18:51
I tend to agree, the question should better be moved. –  Gerald Feb 10 at 20:15
This question appears to be off-topic because it is about physics –  Eduardo Serra Feb 10 at 21:27
How do I move it? –  frodeborli Feb 11 at 9:49

Let's restrict to special relativity, meaning two inertial frames moving in a Minkowski spacetime.

A clock in the first inertial frame ticks slower, when seen from the second. A clock in the second inertial frames ticks slower, when seen from the first.

Now assume, that you are fixed to one of the two inertial frames. Usually you measure velocities within your own inertial frame, meaning measuring distance and time in your inertial frame to calculate the velocity of the other frame. In this case you'll get a velocity below the speed of light.

If you measure the distance in your inertial frame, and divide this distance by the time you observe on the clock of the moving inertial frame, you'll get higher velocities, which may exceed the speed of light. But this result has only the dimension of a velocity; it is not a velocity with respect of any of the two inertial frames.

The moving observer would observe a slow-down of external events, not a speed-up. The observer would observe a speed-up only in the sense of a relativistic Doppler blue-shift when moving towards the observed object.

The people on the destination planet age the same as on the Earth, provided they don't move relative to the Earth. It's just you, since you travel that fast, the distance gets shorter, and therefore the time for the travel also gets shorter. By the Doppler shift and space contraction you'll see them age faster.

From Alpha Centauri, it's again the space contraction for the traveller, making the distance shorter. Again we have a combined space contraction/Doppler effect looking the people on Earth aging rapidly for about 4.5 years.

For people on Earth observing the travel takes 4.5 years. For the traveller it may take just a few days. With an acceleration of just 1g you could traverse the Milky Way in about 20 years, seen from the spaceship. The same travel seen from Earth would take about 100,000 years.

-
So, considering a clock starting travel today from earth, traveling to Alpha Centauri at 99.9% the speed of light. After about 4.5 years it arrives and stops. How much time would have passed on this clock? I assume that the clock will have advanced somewhere near 40 hours - making the speed of light for that clock seem to be (4.5 light years) / (40 hours) = 2.95*10^11 m/s. –  frodeborli Feb 10 at 14:52
To clarify; I think I am trying to use local time on proper distance. –  frodeborli Feb 10 at 15:08
It would be 43.4 days for the clock: 4.5 years x $\sqrt{1-0.999^2}$ –  Gerald Feb 10 at 17:27
The distance for the clock would contract the same factor making the distance only 43.4 light days. The speed would still be 99.9% the speed of light. –  Gerald Feb 10 at 17:31
@self. That's the twin paradox (actually no paradox), the clue is the change of inertial frames due to acceleration/deceleration: csupomona.edu/~ajm/materials/twinparadox.html –  Gerald Feb 11 at 11:10

One is always traveling. Even if traveling at the speed of light, one is still affected by the forces. If the clock depends on gravity, then inertia will have an affect on it. If the clock uses a battery, then maybe the clock will only affect itself if nothing else is nearby.

I'm not an expert, but even at the speed of light, forces will still have an effect.

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# Runge-Kutta Method and Stability Function

consider a Runge-Kutta method with $$R$$ stages and coefficients given by Butcher-Tableau:

$$\begin{array}{c|c} a & B \\ \hline & c^T \\ \end{array}$$

Now consider the step $$t \rightarrow t + h, y_0 \rightarrow y_1$$ and let $$g_i = y_0 + h \sum_{i=0}^R B_{i,j}k_j$$. So the $$g_i$$ are approximations of $$u(t+a_ih)$$ and the $$k_i$$ are approximations of $$u'(t+a_ih)$$.

I have some questions concerning this construction:

1. How do I get (probably implicit) formulas for $$g = (g_1, ..., g_R)^T$$ and $$y_1$$.

2. Can I then use them to get $$\omega(z) = y_1 = \frac{det(Id-zB-zec^T)}{det(Id-zB)}$$ ($$\omega$$ is the stability function)? Maybe by applying the formulas of 1. to $$u'(t) = \lambda u(t), u(0) = 1$$?

• To say nothing against the variability of variable-letter assignment, but the usual convention for Butcher tableaux has them as $\begin{array}{c|c}c&A\\\hline&b^T\end{array}$. Dec 3, 2019 at 15:22
• @Dr.LutzLehmann Yes that's true, but unfortunately we do it differently in our lecture. Dec 3, 2019 at 15:27

2. Yes, the stability function covers the behavior for linear problems, with a focus on non-expanding dynamics. You get to solve $$\newcommand{\ones}{{\bf 1}}$$ $$k=λ(y\ones+hBk)\implies k=λ(I-λhB)^{-1}\ones y$$ and $$y_{+1}=y+hc^Tλ(I-λhB)^{-1}y=(1+(λh)c^T(I-λhB)^{-1}\ones)y$$.
Then apply the determinant identity $$\det(I+uv^T)=1+v^Tu$$ in reverse $$1+(λh)c^T(I-λhB)^{-1}\ones=\det(I+(λh)\ones c^T(I-λhB)^{-1})=\frac{\det(I-λhB+(λh)\ones c^T)}{\det(I-λhB)}$$
• Newton's method or similar. In practice, one would prefer to evaluate the Jacobian at most once per step or use the same over several steps. With the corresponding simplified Newton method you get linear convergence with a rate $O(h^2)$ which is still faster than the "naive" fixed-point iteration of the stage equations which has a rate $O(h)$. Dec 3, 2019 at 16:32


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## Kumar, Devendra

Compute Distance To:
 Author ID: kumar.devendra.3 Published as: Kumar, Devendra; Kumar, D. External Links: ORCID
 Documents Indexed: 165 Publications since 2010 5 Contributions as Editor Co-Authors: 62 Co-Authors with 108 Joint Publications 3,295 Co-Co-Authors
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### Serials

 8 Mathematical Methods in the Applied Sciences 7 Applications and Applied Mathematics 6 Chaos, Solitons and Fractals 6 Advances in Difference Equations 6 AIMS Mathematics 5 Mathematical Modelling of Natural Phenomena 5 Discrete and Continuous Dynamical Systems. Series S 4 Applied Mathematics and Computation 4 Journal of Computational and Applied Mathematics 4 International Journal of Computer Mathematics 4 International Journal of Biomathematics 4 International Journal of Applied and Computational Mathematics 3 Physica A 3 Journal of the Egyptian Mathematical Society 3 Abstract and Applied Analysis 3 Journal of Applied Mathematics and Computing 3 Journal of Applied Mathematics, Statistics and Informatics 3 Mathematics and Its Applications: Modelling, Engineering, and Social Sciences 2 Mathematics and Computers in Simulation 2 Applied Numerical Mathematics 2 Numerical Methods for Partial Differential Equations 2 The Journal of Analysis 2 Computational and Applied Mathematics 2 Journal of Applied Analysis 2 Nonlinear Dynamics 2 Chaos 2 Proceedings of the National Academy of Sciences, India. Section A. Physical Sciences 2 International Journal of Mathematics and Computer Science 2 International Journal of Nonlinear Science 2 Journal of Fractional Calculus and Applications 2 International Journal of Analysis and Applications 2 Mathematical Sciences & Applications E-Notes 2 Journal of Numerical Analysis and Approximation Theory 2 Waves, Wavelets and Fractals. Advanced Analysis 2 Stochastics and Quality Control 1 International Journal of Modern Physics B 1 Archives of Mechanics 1 Journal of Mathematical Analysis and Applications 1 Journal of the Korean Mathematical Society 1 Kyungpook Mathematical Journal 1 Le Matematiche 1 Rendiconti del Circolo Matemàtico di Palermo. Serie II 1 Journal of the Nigerian Mathematical Society 1 Applied Mathematical Modelling 1 Communications in Statistics. Simulation and Computation 1 Communications in Statistics. Theory and Methods 1 Filomat 1 Complexity 1 Journal of Difference Equations and Applications 1 Journal of Mathematical Chemistry 1 Journal of Applied Statistics 1 Analysis (München) 1 Far East Journal of Mathematical Sciences 1 International Journal of Nonlinear Sciences and Numerical Simulation 1 Kragujevac Journal of Mathematics 1 Hacettepe Journal of Mathematics and Statistics 1 Annali dell’Università di Ferrara. Sezione VII. Scienze Matematiche 1 International Mathematical Forum 1 Scientia. Series A: Mathematical Sciences. New Series 1 Applied Mathematical Sciences (Ruse) 1 European Journal of Pure and Applied Mathematics 1 Advanced Studies in Theoretical Physics 1 Numerical Mathematics: Theory, Methods and Applications 1 East Asian Mathematical Journal 1 Journal of Nonlinear Science and Applications 1 Advances in Theoretical and Applied Mechanics 1 Acta Universitatis Sapientiae. Mathematica 1 Bulletin of Mathematical Analysis and Applications 1 Journal of Probability and Statistics 1 Journal of Applied Mathematics & Informatics 1 Azerbaijan Journal of Mathematics 1 Afrika Matematika 1 Theory and Applications of Mathematics & Computer Science 1 Tamsui Oxford Journal of Information and Mathematical Sciences 1 ISRN Mathematical Physics 1 Springer Proceedings in Mathematics & Statistics 1 Communications in Mathematics and Statistics 1 Advances in Nonlinear Analysis 1 Journal of Complex Analysis 1 Gulf Journal of Mathematics 1 Journal of Function Spaces 1 Studies in Systems, Decision and Control 1 International Journal of Advances in Applied Mathematics and Mechanics 1 Results in Applied Mathematics
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 57 Partial differential equations (35-XX) 46 Numerical analysis (65-XX) 38 Real functions (26-XX) 30 Ordinary differential equations (34-XX) 29 Special functions (33-XX) 24 Biology and other natural sciences (92-XX) 23 Integral transforms, operational calculus (44-XX) 17 Statistics (62-XX) 9 Functions of a complex variable (30-XX) 9 Probability theory and stochastic processes (60-XX) 8 Fluid mechanics (76-XX) 7 Approximations and expansions (41-XX) 6 Statistical mechanics, structure of matter (82-XX) 5 General and overarching topics; collections (00-XX) 5 Several complex variables and analytic spaces (32-XX) 4 Computer science (68-XX) 4 Mechanics of deformable solids (74-XX) 3 Integral equations (45-XX) 3 Operator theory (47-XX) 3 Astronomy and astrophysics (85-XX) 2 Potential theory (31-XX) 2 Dynamical systems and ergodic theory (37-XX) 2 Harmonic analysis on Euclidean spaces (42-XX) 2 Classical thermodynamics, heat transfer (80-XX) 2 Systems theory; control (93-XX) 1 Mathematical logic and foundations (03-XX) 1 Number theory (11-XX) 1 Quantum theory (81-XX) 1 Geophysics (86-XX) 1 Information and communication theory, circuits (94-XX)

### Citations contained in zbMATH Open

96 Publications have been cited 1,059 times in 583 Documents Cited by Year
A fractional epidemiological model for computer viruses pertaining to a new fractional derivative. Zbl 1426.68015
Singh, Jagdev; Kumar, Devendra; Hammouch, Zakia; Atangana, Abdon
2018
A new fractional model for giving up smoking dynamics. Zbl 1422.34062
Singh, Jagdev; Kumar, Devendra; Al Qurashi, Maysaa; Baleanu, Dumitru
2017
A modified numerical scheme and convergence analysis for fractional model of Lienard’s equation. Zbl 1404.34007
Kumar, Devendra; Agarwal, Ravi P.; Singh, Jagdev
2018
New aspects of fractional Biswas-Milovic model with Mittag-Leffler law. Zbl 1423.35415
Singh, Jagdev; Kumar, Devendra; Baleanu, Dumitru
2019
A new analysis for fractional model of regularized long-wave equation arising in ion acoustic plasma waves. Zbl 1388.35212
Kumar, Devendra; Singh, Jagdev; Baleanu, Dumitru
2017
On the analysis of vibration equation involving a fractional derivative with Mittag-Leffler law. Zbl 1442.35515
Kumar, Devendra; Singh, Jagdev; Baleanu, Dumitru
2020
A new numerical algorithm for fractional Fitzhugh-Nagumo equation arising in transmission of nerve impulses. Zbl 1390.35376
Kumar, Devendra; Singh, Jagdev; Baleanu, Dumitru
2018
An efficient analytical technique for fractional model of vibration equation. Zbl 1446.74057
Srivastava, H. M.; Kumar, Devendra; Singh, Jagdev
2017
A new analysis of fractional fish farm model associated with Mittag-Leffler-type kernel. Zbl 1442.34131
Singh, Jagdev; Kumar, Devendra; Baleanu, Dumitru
2020
On the analysis of fractional diabetes model with exponential law. Zbl 1446.34018
Singh, Jagdev; Kumar, Devendra; Baleanu, Dumitru
2018
A hybrid computational approach for Klein-Gordon equations on Cantor sets. Zbl 1371.35326
Kumar, Devendra; Singh, Jagdev; Baleanu, Dumitru
2017
On the local fractional wave equation in fractal strings. Zbl 1419.35226
Singh, Jagdev; Kumar, Devendra; Baleanu, Dumitru; Rathore, Sushila
2019
Analysis of an El Nino-Southern Oscillation model with a new fractional derivative. Zbl 1373.86007
Singh, Jagdev; Kumar, Devendra; Nieto, Juan J.
2017
An efficient numerical algorithm for the fractional Drinfeld-Sokolov-Wilson equation. Zbl 1427.65324
Singh, Jagdev; Kumar, Devendra; Baleanu, Dumitru; Rathore, Sushila
2018
Fractional modelling arising in unidirectional propagation of long waves in dispersive media. Zbl 1431.35230
Kumar, Sunil; Kumar, Devendra; Singh, Jagdev
2016
On the analysis of chemical kinetics system pertaining to a fractional derivative with Mittag-Leffler type kernel. Zbl 1390.34027
Singh, Jagdev; Kumar, Devendra; Baleanu, Dumitru
2017
Homotopy perturbation method for fractional gas dynamics equation using Sumudu transform. Zbl 1262.76082
Singh, Jagdev; Kumar, Devendra; Kılıçman, A.
2013
Numerical solution of predator-prey model with Beddington-DeAngelis functional response and fractional derivatives with Mittag-Leffler kernel. Zbl 1421.34033
2019
A reliable numerical algorithm for the fractional vibration equation. Zbl 1380.65207
Singh, Harendra; Srivastava, H. M.; Kumar, Devendra
2017
A hybrid analytical algorithm for nonlinear fractional wave-like equations. Zbl 1423.65001
Kumar, Devendra; Singh, Jagdev; Purohit, Sunil Dutt; Swroop, Ram
2019
Numerical computation of nonlinear fractional Zakharov-Kuznetsov equation arising in ion-acoustic waves. Zbl 06363304
Kumar, Devendra; Singh, Jagdev; Kumar, Sunil
2014
Homotopy perturbation Sumudu transform method for nonlinear equations. Zbl 1247.76062
Singh, Jagdev; Kumar, Devendra; Sushila
2011
An efficient technique for nonlinear time-fractional Klein-Fock-Gordon equation. Zbl 1433.35454
Veeresha, P.; Prakasha, D. G.; Kumar, Devendra
2020
An efficient approach for fractional Harry Dym equation by using Sumudu transform. Zbl 1275.65086
Kumar, Devendra; Singh, Jagdev; Kılıçman, A.
2013
Analytical solution of Abel integral equation arising in astrophysics via Laplace transform. Zbl 1311.45001
Kumar, Sunil; Kumar, Amit; Kumar, Devendra; Singh, Jagdev; Singh, Arvind
2015
Numerical study for systems of fractional differential equations via Laplace transform. Zbl 1330.65127
Gupta, Sumit; Kumar, Devendra; Singh, Jagdev
2015
Sumudu decomposition method for nonlinear equations. Zbl 1250.65126
Kumar, Devendra; Singh, Jagdev; Rathore, Sushila
2012
Numerical solutions of nonlinear fractional partial differential equations arising in spatial diffusion of biological populations. Zbl 1474.65415
Singh, Jagdev; Kumar, Devendra; Kılıçman, Adem
2014
A study of Marichev-Saigo-Maeda fractional integral operators associated with the S-generalized Gauss hypergeometric function. Zbl 1434.33012
Bansal, Manish Kumar; Kumar, Devendra; Jain, Rashmi
2019
Exponentiated Chen distribution: properties and estimation. Zbl 1383.62037
Dey, Sanku; Kumar, Devendra; Ramos, Pedro L.; Louzada, Francisco
2017
Alpha power transformed inverse Lindley distribution: a distribution with an upside-down bathtub-shaped hazard function. Zbl 1404.60028
Dey, Sanku; Nassar, Mazen; Kumar, Devendra
2019
An efficient computational technique for local fractional heat conduction equations in fractal media. Zbl 1412.35374
Zhao, Duan; Singh, Jagdev; Kumar, Devendra; Rathore, Sushila; Yang, Xiao-Jun
2017
An efficient computational method for local fractional transport equation occurring in fractal porous media. Zbl 1463.76050
Singh, Jagdev; Kumar, Devendra; Kumar, Sunil
2020
A new fractional SIRS-SI malaria disease model with application of vaccines, antimalarial drugs, and spraying. Zbl 1485.92136
Kumar, Devendra; Singh, Jagdev; Al Qurashi, Maysaa; Baleanu, Dumitru
2019
Interrelationships between Marichev-Saigo-Maeda fractional integral operators, the Laplace transform and the $$\overline{H}$$-function. Zbl 1418.33001
Bansal, M. K.; Kumar, Devendra; Jain, R.
2019
A reliable algorithm for the approximate solution of the nonlinear Lane-Emden type equations arising in astrophysics. Zbl 1407.85001
Singh, Harendra; Srivastava, H. M.; Kumar, Devendra
2018
Extended exponential distribution based on order statistics. Zbl 1377.62135
Kumar, Devendra; Dey, Sanku; Nadarajah, Saralees
2017
A new extension of Weibull distribution: properties and different methods of estimation. Zbl 1390.60062
Nassar, Mazen; Afify, Ahmed Z.; Dey, Sanku; Kumar, Devendra
2018
A parameter-uniform numerical scheme for the parabolic singularly perturbed initial boundary value problems with large time delay. Zbl 1418.65101
Kumar, Devendra; Kumari, Parvin
2019
An implicit scheme for singularly perturbed parabolic problem with retarded terms arising in computational neuroscience. Zbl 1407.65111
Kumar, Devendra
2018
Certain fractional calculus and integral transform results of incomplete $$\aleph$$-functions with applications. Zbl 1445.26002
Bansal, Manish Kumar; Kumar, Devendra; Nisar, K. S.; Singh, Jagdev
2020
Application of He’s homotopy perturbation method for solving nonlinear wave-like equations with variable coefficients. Zbl 1360.35115
Gupta, Sumit; Kumar, Devendra; Singh, Jagdev
2013
On the solutions of generalized fractional kinetic equations. Zbl 1227.82055
Chaurasia, V. B. L.; Kumar, Devendra
2010
Analytical techniques for system of time fractional nonlinear differential equations. Zbl 1370.34010
Choi, Junesang; Kumar, Devendra; Singh, Jagdev; Swroop, Ram
2017
On the solutions of fractional reaction-diffusion equations. Zbl 1282.35402
Singh, Jagdev; Kumar, Devendra; Rathore, Sushila
2013
On the distribution of mixed sum of independent random variables one of them associated with Srivastava’s polynomials and $$\overline{H}$$-function. Zbl 1292.62033
Singh, Jagdev; Kumar, Devendra
2014
An integral operator involving generalized Mittag-Leffler function and associated fractional calculus results. Zbl 1426.33052
Bansal, M. K.; Jolly, N.; Jain, R.; Kumar, Devendra
2019
Parameter-uniform numerical treatment of singularly perturbed initial-boundary value problems with large delay. Zbl 1436.35028
Kumar, Devendra; Kumari, Parvin
2020
A parameter-uniform method for singularly perturbed turning point problems exhibiting interior or twin boundary layers. Zbl 07474845
Kumar, Devendra
2019
Homotopy perturbation algorithm using Laplace transform for gas dynamics equation. Zbl 1277.65085
Singh, Jagdev; Kumar, Devendra; Sushila
2012
A collocation method for singularly perturbed differential-difference turning point problems exhibiting boundary/interior layers. Zbl 1405.65081
Kumar, Devendra
2018
Analytic study of Allen-Cahn equation of fractional order. Zbl 1409.35220
Kumar, Devendra; Singh, Jagdev; Baleanu, Dumitru
2017
New aspects of fractional Bloch model associated with composite fractional derivative. Zbl 1469.82039
Singh, Jagdev; Kumar, Devendra; Baleanu, Dumitru
2021
Certain formulas involving a multi-index Mittag-Leffler function. Zbl 1429.33033
Bansal, Manish Kumar; Harjule, P.; Choi, Junesang; Mubeen, Shahid; Kumar, Devendra
2019
Statistical inference of exponentiated moment exponential distribution based on lower record values. Zbl 1380.62065
Kumar, Devendra; Dey, Tanujit; Dey, Sanku
2017
Quadratic $$B$$-spline collocation method for two-parameter singularly perturbed problem on exponentially graded mesh. Zbl 1480.65187
Shivhare, Meenakshi; Pramod Chakravarthy, P.; Kumar, Devendra
2021
The Marshall-Olkin alpha power family of distributions with applications. Zbl 1409.62047
Nassar, Mazen; Kumar, Devendra; Dey, Sanku; Cordeiro, Gauss M.; Afify, Ahmed Z.
2019
On the solutions of integral equations of Fredholm type with special functions. Zbl 1275.33033
Chaurasia, V. B. L.; Kumar, Devendra
2012
A uniformly convergent quadratic B-spline collocation method for singularly perturbed parabolic partial differential equations with two small parameters. Zbl 1471.65164
Shivhare, Meenakshi; Podila, Pramod Chakravarthy; Kumar, Devendra
2021
Higher order B-spline differential quadrature rule to approximate generalized Rosenau-RLW equation. Zbl 1451.65157
Mustahsan, Muhammad; Kiran, Ayesha; Singh, Jagdev; Nisar, Kottakkaran Sooppy; Kumar, Devendra
2020
New aspects of fractional epidemiological model for computer viruses with Mittag-Leffler law. Zbl 07357307
Kumar, Devendra; Singh, Jagdev
2020
Fractional SIR epidemic model of childhood disease with Mittag-Leffler memory. Zbl 1466.92210
Veeresha, P.; Prakasha, D. G.; Kumar, Devendra
2021
A reliable analytical algorithm for cubic isothermal auto-catalytic chemical system. Zbl 1429.65265
Saad, Khaled M.; Srivastava, H. M.; Kumar, Devendra
2019
The integration of certain product involving special functions. Zbl 1244.26009
Chaurasia, V. B. L.; Kumar, Devendra
2010
Fitted mesh method for a class of singularly perturbed differential-difference equations. Zbl 1349.65203
Kumar, Devendra
2015
Generalized growth and approximation of entire function solution of Helmholtz equation in Banach spaces. Zbl 1341.30026
Kumar, Devendra
2016
On the growth and best polynomial approximation of entire functions in $$\mathbb {C}^{m}$$ ($$m\geq 2$$) in some Banach spaces. Zbl 1371.32004
Kumar, Devendra
2017
Lie symmetry analysis and dynamical structures of soliton solutions for the $$(2+1)$$-dimensional modified CBS equation. Zbl 1451.35167
Kumar, S.; Kumar, D.
2020
Solutions of unified fractional Schrödinger equations. Zbl 1234.35296
Chaurasia, V. B. L.; Kumar, Devendra
2012
Methods of mathematical modelling. Fractional differential equations. Zbl 1425.00088
2020
Fractional calculus in medical and health science. Zbl 1460.92004
2021
Bounded M-O extended exponential distribution with applications. Zbl 1448.60032
Ghosh, Indranil; Dey, Sanku; Kumar, Devendra
2019
An efficient computational approach for linear and nonlinear fractional differential equations. Zbl 1432.65116
Singh, Jagdev; Kumar, Devendra; Swroop, Ram; Sharma, Ram Prakash
2017
An efficient computational approach for generalized Hirota-Satsuma coupled KdV equations arising in shallow water waves. Zbl 1431.35154
Gupta, Sumit; Kumar, Devendra; Singh, Jagdev
2017
On the growth and approximation of transcendental entire functions on algebraic varieties. Zbl 1386.41003
Kumar, Devendra
2016
Solution of nonlinear differential equation and special functions. Zbl 1448.33014
Singh, Jagdev; Kumar, Devendra; Bansal, Manish Kumar
2020
Chebyshev type inequalities involving the fractional integral operator containing multi-index Mittag-Leffler function in the kernel. Zbl 1447.26006
Purohit, S. D.; Jolly, N.; Bansal, M. K.; Singh, Jagdev; Kumar, Devendra
2020
Applications of homotopy perturbation transform method for solving time-dependent functional differential equations. Zbl 1394.35076
Gupta, Sumit; Singh, Jagdev; Kumar, Devendra
2013
Application of homotopy perturbation transform method to linear and nonlinear Schrödinger equations. Zbl 1394.65121
Kumar, Devendra; Singh, Jagdev; Sushila
2013
$$L_p$$-approximation of analytic functions on compact sets bounded by Jordan curves. Zbl 1401.30040
Kumar, Devendra; Jain, Vandna
2015
Approximation of entire transcendental functions of several complex variables in some Banach spaces for slow growth. Zbl 1371.32003
Kumar, Devendra
2017
Some fractional calculus results pertaining to Mittag-Leffler type functions. Zbl 1365.26004
Choudhary, Anupama; Kumar, Devendra; Singh, Jagdev
2017
ADMP: a Maple package for symbolic computation and error estimating to singular two-point boundary value problems with initial conditions. Zbl 1451.65130
Gupta, Sumit; Kumar, Devendra; Singh, Jagdev
2019
A family of fractional integrals pertaining to multivariable I-function. Zbl 1207.26012
Chaurasia, V. B. L.; Kumar, Devendra
2010
An efficient numerical method for fractional model of allelopathic stimulatory phytoplankton species with Mittag-Leffler law. Zbl 07396816
Ghanbari, Behzad; Kumar, Devendra; Singh, Jagdev
2021
The type I generalized half-logistic distribution based on upper record values. Zbl 1426.62063
Kumar, Devendra; Jain, Neetu; Gupta, Shivani
2015
Numerical solution of time-fractional three-species food chain model arising in the realm of mathematical ecology. Zbl 1443.92192
Dubey, Ved Prakash; Kumar, Rajnesh; Kumar, Devendra
2020
On the Elzaki transform and its applications in fractional free electron laser equation. Zbl 1450.44004
Singh, Yudhveer; Gill, Vinod; Kundu, Sunil; Kumar, Devendra
2019
Moments and estimation of reduced Kies distribution based on progressive type-II right censored order statistics. Zbl 1471.62282
Dey, Sanku; Nassar, Mazen; Kumar, Devendra
2019
A reliable numerical approach for nonlinear fractional optimal control problems. Zbl 07412521
Singh, Harendra; Pandey, Rajesh K.; Kumar, Devendra
2021
Alternative derivation of generalized fractional kinetic equations. Zbl 07441447
Saxena, R. K.; Ram, J.; Kumar, D.
2013
A comparative analysis of two computational schemes for solving local fractional Laplace equations. Zbl 07442127
Dubey, Ved Prakash; Singh, Jagdev; Alshehri, Ahmed M.; Dubey, Sarvesh; Kumar, Devendra
2021
Analytical approach for fractional extended Fisher-Kolmogorov equation with Mittag-Leffler kernel. Zbl 1482.35257
Veeresha, P.; Prakasha, D. G.; Singh, Jagdev; Khan, Ilyas; Kumar, Devendra
2020
Trigonometric quintic $$B$$-spline collocation method for singularly perturbed turning point boundary value problems. Zbl 1480.65177
2021
A hybrid analytical scheme for the numerical computation of time fractional computer virus propagation model and its stability analysis. Zbl 1483.68022
Dubey, Ved Prakash; Kumar, Rajnesh; Kumar, Devendra
2020
Identifying the space source term problem for time-space-fractional diffusion equation. Zbl 07507523
Karapinar, Erdal; Kumar, Devendra; Sakthivel, Rathinasamy; Luc, Nguyen Hoang; Can, N. H.
2020
New aspects of fractional Bloch model associated with composite fractional derivative. Zbl 1469.82039
Singh, Jagdev; Kumar, Devendra; Baleanu, Dumitru
2021
Quadratic $$B$$-spline collocation method for two-parameter singularly perturbed problem on exponentially graded mesh. Zbl 1480.65187
Shivhare, Meenakshi; Pramod Chakravarthy, P.; Kumar, Devendra
2021
A uniformly convergent quadratic B-spline collocation method for singularly perturbed parabolic partial differential equations with two small parameters. Zbl 1471.65164
Shivhare, Meenakshi; Podila, Pramod Chakravarthy; Kumar, Devendra
2021
Fractional SIR epidemic model of childhood disease with Mittag-Leffler memory. Zbl 1466.92210
Veeresha, P.; Prakasha, D. G.; Kumar, Devendra
2021
Fractional calculus in medical and health science. Zbl 1460.92004
2021
An efficient numerical method for fractional model of allelopathic stimulatory phytoplankton species with Mittag-Leffler law. Zbl 07396816
Ghanbari, Behzad; Kumar, Devendra; Singh, Jagdev
2021
A reliable numerical approach for nonlinear fractional optimal control problems. Zbl 07412521
Singh, Harendra; Pandey, Rajesh K.; Kumar, Devendra
2021
A comparative analysis of two computational schemes for solving local fractional Laplace equations. Zbl 07442127
Dubey, Ved Prakash; Singh, Jagdev; Alshehri, Ahmed M.; Dubey, Sarvesh; Kumar, Devendra
2021
Trigonometric quintic $$B$$-spline collocation method for singularly perturbed turning point boundary value problems. Zbl 1480.65177
2021
On the analysis of vibration equation involving a fractional derivative with Mittag-Leffler law. Zbl 1442.35515
Kumar, Devendra; Singh, Jagdev; Baleanu, Dumitru
2020
A new analysis of fractional fish farm model associated with Mittag-Leffler-type kernel. Zbl 1442.34131
Singh, Jagdev; Kumar, Devendra; Baleanu, Dumitru
2020
An efficient technique for nonlinear time-fractional Klein-Fock-Gordon equation. Zbl 1433.35454
Veeresha, P.; Prakasha, D. G.; Kumar, Devendra
2020
An efficient computational method for local fractional transport equation occurring in fractal porous media. Zbl 1463.76050
Singh, Jagdev; Kumar, Devendra; Kumar, Sunil
2020
Certain fractional calculus and integral transform results of incomplete $$\aleph$$-functions with applications. Zbl 1445.26002
Bansal, Manish Kumar; Kumar, Devendra; Nisar, K. S.; Singh, Jagdev
2020
Parameter-uniform numerical treatment of singularly perturbed initial-boundary value problems with large delay. Zbl 1436.35028
Kumar, Devendra; Kumari, Parvin
2020
Higher order B-spline differential quadrature rule to approximate generalized Rosenau-RLW equation. Zbl 1451.65157
Mustahsan, Muhammad; Kiran, Ayesha; Singh, Jagdev; Nisar, Kottakkaran Sooppy; Kumar, Devendra
2020
New aspects of fractional epidemiological model for computer viruses with Mittag-Leffler law. Zbl 07357307
Kumar, Devendra; Singh, Jagdev
2020
Lie symmetry analysis and dynamical structures of soliton solutions for the $$(2+1)$$-dimensional modified CBS equation. Zbl 1451.35167
Kumar, S.; Kumar, D.
2020
Methods of mathematical modelling. Fractional differential equations. Zbl 1425.00088
2020
Solution of nonlinear differential equation and special functions. Zbl 1448.33014
Singh, Jagdev; Kumar, Devendra; Bansal, Manish Kumar
2020
Chebyshev type inequalities involving the fractional integral operator containing multi-index Mittag-Leffler function in the kernel. Zbl 1447.26006
Purohit, S. D.; Jolly, N.; Bansal, M. K.; Singh, Jagdev; Kumar, Devendra
2020
Numerical solution of time-fractional three-species food chain model arising in the realm of mathematical ecology. Zbl 1443.92192
Dubey, Ved Prakash; Kumar, Rajnesh; Kumar, Devendra
2020
Analytical approach for fractional extended Fisher-Kolmogorov equation with Mittag-Leffler kernel. Zbl 1482.35257
Veeresha, P.; Prakasha, D. G.; Singh, Jagdev; Khan, Ilyas; Kumar, Devendra
2020
A hybrid analytical scheme for the numerical computation of time fractional computer virus propagation model and its stability analysis. Zbl 1483.68022
Dubey, Ved Prakash; Kumar, Rajnesh; Kumar, Devendra
2020
Identifying the space source term problem for time-space-fractional diffusion equation. Zbl 07507523
Karapinar, Erdal; Kumar, Devendra; Sakthivel, Rathinasamy; Luc, Nguyen Hoang; Can, N. H.
2020
New aspects of fractional Biswas-Milovic model with Mittag-Leffler law. Zbl 1423.35415
Singh, Jagdev; Kumar, Devendra; Baleanu, Dumitru
2019
On the local fractional wave equation in fractal strings. Zbl 1419.35226
Singh, Jagdev; Kumar, Devendra; Baleanu, Dumitru; Rathore, Sushila
2019
Numerical solution of predator-prey model with Beddington-DeAngelis functional response and fractional derivatives with Mittag-Leffler kernel. Zbl 1421.34033
2019
A hybrid analytical algorithm for nonlinear fractional wave-like equations. Zbl 1423.65001
Kumar, Devendra; Singh, Jagdev; Purohit, Sunil Dutt; Swroop, Ram
2019
A study of Marichev-Saigo-Maeda fractional integral operators associated with the S-generalized Gauss hypergeometric function. Zbl 1434.33012
Bansal, Manish Kumar; Kumar, Devendra; Jain, Rashmi
2019
Alpha power transformed inverse Lindley distribution: a distribution with an upside-down bathtub-shaped hazard function. Zbl 1404.60028
Dey, Sanku; Nassar, Mazen; Kumar, Devendra
2019
A new fractional SIRS-SI malaria disease model with application of vaccines, antimalarial drugs, and spraying. Zbl 1485.92136
Kumar, Devendra; Singh, Jagdev; Al Qurashi, Maysaa; Baleanu, Dumitru
2019
Interrelationships between Marichev-Saigo-Maeda fractional integral operators, the Laplace transform and the $$\overline{H}$$-function. Zbl 1418.33001
Bansal, M. K.; Kumar, Devendra; Jain, R.
2019
A parameter-uniform numerical scheme for the parabolic singularly perturbed initial boundary value problems with large time delay. Zbl 1418.65101
Kumar, Devendra; Kumari, Parvin
2019
An integral operator involving generalized Mittag-Leffler function and associated fractional calculus results. Zbl 1426.33052
Bansal, M. K.; Jolly, N.; Jain, R.; Kumar, Devendra
2019
A parameter-uniform method for singularly perturbed turning point problems exhibiting interior or twin boundary layers. Zbl 07474845
Kumar, Devendra
2019
Certain formulas involving a multi-index Mittag-Leffler function. Zbl 1429.33033
Bansal, Manish Kumar; Harjule, P.; Choi, Junesang; Mubeen, Shahid; Kumar, Devendra
2019
The Marshall-Olkin alpha power family of distributions with applications. Zbl 1409.62047
Nassar, Mazen; Kumar, Devendra; Dey, Sanku; Cordeiro, Gauss M.; Afify, Ahmed Z.
2019
A reliable analytical algorithm for cubic isothermal auto-catalytic chemical system. Zbl 1429.65265
Saad, Khaled M.; Srivastava, H. M.; Kumar, Devendra
2019
Bounded M-O extended exponential distribution with applications. Zbl 1448.60032
Ghosh, Indranil; Dey, Sanku; Kumar, Devendra
2019
ADMP: a Maple package for symbolic computation and error estimating to singular two-point boundary value problems with initial conditions. Zbl 1451.65130
Gupta, Sumit; Kumar, Devendra; Singh, Jagdev
2019
On the Elzaki transform and its applications in fractional free electron laser equation. Zbl 1450.44004
Singh, Yudhveer; Gill, Vinod; Kundu, Sunil; Kumar, Devendra
2019
Moments and estimation of reduced Kies distribution based on progressive type-II right censored order statistics. Zbl 1471.62282
Dey, Sanku; Nassar, Mazen; Kumar, Devendra
2019
A fractional epidemiological model for computer viruses pertaining to a new fractional derivative. Zbl 1426.68015
Singh, Jagdev; Kumar, Devendra; Hammouch, Zakia; Atangana, Abdon
2018
A modified numerical scheme and convergence analysis for fractional model of Lienard’s equation. Zbl 1404.34007
Kumar, Devendra; Agarwal, Ravi P.; Singh, Jagdev
2018
A new numerical algorithm for fractional Fitzhugh-Nagumo equation arising in transmission of nerve impulses. Zbl 1390.35376
Kumar, Devendra; Singh, Jagdev; Baleanu, Dumitru
2018
On the analysis of fractional diabetes model with exponential law. Zbl 1446.34018
Singh, Jagdev; Kumar, Devendra; Baleanu, Dumitru
2018
An efficient numerical algorithm for the fractional Drinfeld-Sokolov-Wilson equation. Zbl 1427.65324
Singh, Jagdev; Kumar, Devendra; Baleanu, Dumitru; Rathore, Sushila
2018
A reliable algorithm for the approximate solution of the nonlinear Lane-Emden type equations arising in astrophysics. Zbl 1407.85001
Singh, Harendra; Srivastava, H. M.; Kumar, Devendra
2018
A new extension of Weibull distribution: properties and different methods of estimation. Zbl 1390.60062
Nassar, Mazen; Afify, Ahmed Z.; Dey, Sanku; Kumar, Devendra
2018
An implicit scheme for singularly perturbed parabolic problem with retarded terms arising in computational neuroscience. Zbl 1407.65111
Kumar, Devendra
2018
A collocation method for singularly perturbed differential-difference turning point problems exhibiting boundary/interior layers. Zbl 1405.65081
Kumar, Devendra
2018
A new fractional model for giving up smoking dynamics. Zbl 1422.34062
Singh, Jagdev; Kumar, Devendra; Al Qurashi, Maysaa; Baleanu, Dumitru
2017
A new analysis for fractional model of regularized long-wave equation arising in ion acoustic plasma waves. Zbl 1388.35212
Kumar, Devendra; Singh, Jagdev; Baleanu, Dumitru
2017
An efficient analytical technique for fractional model of vibration equation. Zbl 1446.74057
Srivastava, H. M.; Kumar, Devendra; Singh, Jagdev
2017
A hybrid computational approach for Klein-Gordon equations on Cantor sets. Zbl 1371.35326
Kumar, Devendra; Singh, Jagdev; Baleanu, Dumitru
2017
Analysis of an El Nino-Southern Oscillation model with a new fractional derivative. Zbl 1373.86007
Singh, Jagdev; Kumar, Devendra; Nieto, Juan J.
2017
On the analysis of chemical kinetics system pertaining to a fractional derivative with Mittag-Leffler type kernel. Zbl 1390.34027
Singh, Jagdev; Kumar, Devendra; Baleanu, Dumitru
2017
A reliable numerical algorithm for the fractional vibration equation. Zbl 1380.65207
Singh, Harendra; Srivastava, H. M.; Kumar, Devendra
2017
Exponentiated Chen distribution: properties and estimation. Zbl 1383.62037
Dey, Sanku; Kumar, Devendra; Ramos, Pedro L.; Louzada, Francisco
2017
An efficient computational technique for local fractional heat conduction equations in fractal media. Zbl 1412.35374
Zhao, Duan; Singh, Jagdev; Kumar, Devendra; Rathore, Sushila; Yang, Xiao-Jun
2017
Extended exponential distribution based on order statistics. Zbl 1377.62135
Kumar, Devendra; Dey, Sanku; Nadarajah, Saralees
2017
Analytical techniques for system of time fractional nonlinear differential equations. Zbl 1370.34010
Choi, Junesang; Kumar, Devendra; Singh, Jagdev; Swroop, Ram
2017
Analytic study of Allen-Cahn equation of fractional order. Zbl 1409.35220
Kumar, Devendra; Singh, Jagdev; Baleanu, Dumitru
2017
Statistical inference of exponentiated moment exponential distribution based on lower record values. Zbl 1380.62065
Kumar, Devendra; Dey, Tanujit; Dey, Sanku
2017
On the growth and best polynomial approximation of entire functions in $$\mathbb {C}^{m}$$ ($$m\geq 2$$) in some Banach spaces. Zbl 1371.32004
Kumar, Devendra
2017
An efficient computational approach for linear and nonlinear fractional differential equations. Zbl 1432.65116
Singh, Jagdev; Kumar, Devendra; Swroop, Ram; Sharma, Ram Prakash
2017
An efficient computational approach for generalized Hirota-Satsuma coupled KdV equations arising in shallow water waves. Zbl 1431.35154
Gupta, Sumit; Kumar, Devendra; Singh, Jagdev
2017
Approximation of entire transcendental functions of several complex variables in some Banach spaces for slow growth. Zbl 1371.32003
Kumar, Devendra
2017
Some fractional calculus results pertaining to Mittag-Leffler type functions. Zbl 1365.26004
Choudhary, Anupama; Kumar, Devendra; Singh, Jagdev
2017
Fractional modelling arising in unidirectional propagation of long waves in dispersive media. Zbl 1431.35230
Kumar, Sunil; Kumar, Devendra; Singh, Jagdev
2016
Generalized growth and approximation of entire function solution of Helmholtz equation in Banach spaces. Zbl 1341.30026
Kumar, Devendra
2016
On the growth and approximation of transcendental entire functions on algebraic varieties. Zbl 1386.41003
Kumar, Devendra
2016
Analytical solution of Abel integral equation arising in astrophysics via Laplace transform. Zbl 1311.45001
Kumar, Sunil; Kumar, Amit; Kumar, Devendra; Singh, Jagdev; Singh, Arvind
2015
Numerical study for systems of fractional differential equations via Laplace transform. Zbl 1330.65127
Gupta, Sumit; Kumar, Devendra; Singh, Jagdev
2015
Fitted mesh method for a class of singularly perturbed differential-difference equations. Zbl 1349.65203
Kumar, Devendra
2015
$$L_p$$-approximation of analytic functions on compact sets bounded by Jordan curves. Zbl 1401.30040
Kumar, Devendra; Jain, Vandna
2015
The type I generalized half-logistic distribution based on upper record values. Zbl 1426.62063
Kumar, Devendra; Jain, Neetu; Gupta, Shivani
2015
Numerical computation of nonlinear fractional Zakharov-Kuznetsov equation arising in ion-acoustic waves. Zbl 06363304
Kumar, Devendra; Singh, Jagdev; Kumar, Sunil
2014
Numerical solutions of nonlinear fractional partial differential equations arising in spatial diffusion of biological populations. Zbl 1474.65415
Singh, Jagdev; Kumar, Devendra; Kılıçman, Adem
2014
On the distribution of mixed sum of independent random variables one of them associated with Srivastava’s polynomials and $$\overline{H}$$-function. Zbl 1292.62033
Singh, Jagdev; Kumar, Devendra
2014
Homotopy perturbation method for fractional gas dynamics equation using Sumudu transform. Zbl 1262.76082
Singh, Jagdev; Kumar, Devendra; Kılıçman, A.
2013
An efficient approach for fractional Harry Dym equation by using Sumudu transform. Zbl 1275.65086
Kumar, Devendra; Singh, Jagdev; Kılıçman, A.
2013
Application of He’s homotopy perturbation method for solving nonlinear wave-like equations with variable coefficients. Zbl 1360.35115
Gupta, Sumit; Kumar, Devendra; Singh, Jagdev
2013
On the solutions of fractional reaction-diffusion equations. Zbl 1282.35402
Singh, Jagdev; Kumar, Devendra; Rathore, Sushila
2013
Applications of homotopy perturbation transform method for solving time-dependent functional differential equations. Zbl 1394.35076
Gupta, Sumit; Singh, Jagdev; Kumar, Devendra
2013
Application of homotopy perturbation transform method to linear and nonlinear Schrödinger equations. Zbl 1394.65121
Kumar, Devendra; Singh, Jagdev; Sushila
2013
Alternative derivation of generalized fractional kinetic equations. Zbl 07441447
Saxena, R. K.; Ram, J.; Kumar, D.
2013
Sumudu decomposition method for nonlinear equations. Zbl 1250.65126
Kumar, Devendra; Singh, Jagdev; Rathore, Sushila
2012
Homotopy perturbation algorithm using Laplace transform for gas dynamics equation. Zbl 1277.65085
Singh, Jagdev; Kumar, Devendra; Sushila
2012
On the solutions of integral equations of Fredholm type with special functions. Zbl 1275.33033
Chaurasia, V. B. L.; Kumar, Devendra
2012
Solutions of unified fractional Schrödinger equations. Zbl 1234.35296
Chaurasia, V. B. L.; Kumar, Devendra
2012
Homotopy perturbation Sumudu transform method for nonlinear equations. Zbl 1247.76062
Singh, Jagdev; Kumar, Devendra; Sushila
2011
On the solutions of generalized fractional kinetic equations. Zbl 1227.82055
Chaurasia, V. B. L.; Kumar, Devendra
2010
The integration of certain product involving special functions. Zbl 1244.26009
Chaurasia, V. B. L.; Kumar, Devendra
2010
A family of fractional integrals pertaining to multivariable I-function. Zbl 1207.26012
Chaurasia, V. B. L.; Kumar, Devendra
2010
all top 5

### Cited by 1,039 Authors

 110 Advances in Difference Equations 66 Chaos, Solitons and Fractals 33 Fractals 32 International Journal of Applied and Computational Mathematics 24 AIMS Mathematics 19 Applied Mathematics and Computation 18 Discrete and Continuous Dynamical Systems. Series S 18 Journal of Function Spaces 17 Mathematical Modelling of Natural Phenomena 16 Boundary Value Problems 11 Mathematics and Computers in Simulation 11 Computational and Applied Mathematics 10 Journal of Nonlinear Science and Applications 10 Advances in Mathematical Physics 9 Applied Numerical Mathematics 9 International Journal of Biomathematics 7 Computers & Mathematics with Applications 7 Physica A 7 Journal of Computational and Applied Mathematics 7 Abstract and Applied Analysis 7 Chaos 5 Mathematical Problems in Engineering 5 Applications and Applied Mathematics 4 International Journal of Computer Mathematics 4 International Journal of Nonlinear Sciences and Numerical Simulation 4 Mathematical Sciences 4 Computational Methods for Differential Equations 3 Nonlinear Dynamics 3 International Journal of Differential Equations 3 Journal of Mathematics 2 Journal of Computational Physics 2 Mathematical Methods in the Applied Sciences 2 Kyungpook Mathematical Journal 2 Computational Mathematics and Mathematical Physics 2 The Journal of Analysis 2 Filomat 2 Differential Equations and Dynamical Systems 2 Discrete Dynamics in Nature and Society 2 Qualitative Theory of Dynamical Systems 2 Proceedings of the National Academy of Sciences, India. Section A. Physical Sciences 2 Journal of Applied Mathematics and Computing 2 Mathematical Biosciences and Engineering 2 Waves in Random and Complex Media 2 Acta Universitatis Sapientiae. Mathematica 2 Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A: Matemáticas. RACSAM 2 Palestine Journal of Mathematics 2 Evolution Equations and Control Theory 2 Mathematics 2 International Journal of Advances in Applied Mathematics and Mechanics 1 Acta Mechanica 1 Journal of the Franklin Institute 1 Nonlinearity 1 Physics Letters. A 1 Calcolo 1 International Journal of Mathematics and Mathematical Sciences 1 Semigroup Forum 1 Tamkang Journal of Mathematics 1 Journal of the Nigerian Mathematical Society 1 Applied Mathematics Letters 1 Computational Mathematics and Modeling 1 Numerical Algorithms 1 Applied Mathematical Modelling 1 SIAM Journal on Applied Mathematics 1 Electronic Journal of Differential Equations (EJDE) 1 Journal of the Egyptian Mathematical Society 1 Advances in Computational Mathematics 1 Complexity 1 Boletín de la Sociedad Matemática Mexicana. Third Series 1 Electronic Journal of Probability 1 Journal of Applied Analysis 1 Journal of Inequalities and Applications 1 Fractional Calculus & Applied Analysis 1 Analysis (München) 1 Communications in Nonlinear Science and Numerical Simulation 1 Far East Journal of Mathematical Sciences 1 Nonlinear Functional Analysis and Applications 1 Kragujevac Journal of Mathematics 1 Bulletin of the Brazilian Mathematical Society. New Series 1 Boletim da Sociedade Paranaense de Matemática. Terceira Série 1 International Journal of Computational Methods 1 Mediterranean Journal of Mathematics 1 Journal of Statistical Mechanics: Theory and Experiment 1 The Journal of Prime Research in Mathematics 1 Asian-European Journal of Mathematics 1 Tbilisi Mathematical Journal 1 Advances in Differential Equations and Control Processes 1 Bulletin of Mathematical Analysis and Applications 1 Symmetry 1 Afrika Matematika 1 Journal of Mathematical Extension 1 S$$\vec{\text{e}}$$MA Journal 1 Journal of Applied Analysis and Computation 1 Axioms 1 Electronic Journal of Mathematical Analysis and Applications EJMAA 1 International Journal of Analysis and Applications 1 Journal of Applied Mathematics, Statistics and Informatics 1 Iranian Journal of Numerical Analysis and Optimization 1 Open Mathematics 1 Waves, Wavelets and Fractals. Advanced Analysis


text:
## Practise solving quadratic equations algebraically with this self-marking exercise.

##### FactorisingLevel 1Level 2Level 3Level 4Level 5Level 6Level 7Level 8ExamDescHelp

This is level 1; A quadratic equation presented in a factorised form. You can earn a trophy if you get at least 7 correct.

 $$(x − 3)(x − 7) = 0$$ x = and $$(x + 2)(x + 1) = 0$$ x = and $$(x − 1)(x − 4) = 0$$ x = and $$(x + 3)(x + 1) = 0$$ x = and $$(x − 4)(x − 7) = 0$$ x = and $$(x + 1)(x − 3) = 0$$ x = and $$(2x + 3)(x + 4) = 0$$ x = and $$(4x − 1)(x − 5) = 0$$ x = and $$(2x − 5)(x − 7) = 0$$ x = and $$(2x − 3)(5x − 5) = 0$$ x = and $$(2x + 1)(4x + 3) = 0$$ x = and $$5(2x − 1)(5x − 1) = 0$$ x = and
Check

This is Quadratic Equations level 1. You can also try:
Level 2 Level 3 Level 4 Level 5 Level 6 Level 7 Level 8

## Instructions

Try your best to answer the questions above. Type your answers into the boxes provided leaving no spaces. As you work through the exercise regularly click the "check" button. If you have any wrong answers, do your best to do corrections but if there is anything you don't understand, please ask your teacher for help.

When you have got all of the questions correct you may want to print out this page and paste it into your exercise book. If you keep your work in an ePortfolio you could take a screen shot of your answers and paste that into your Maths file.

## Transum.org

This web site contains over a thousand free mathematical activities for teachers and pupils. Click here to go to the main page which links to all of the resources available.

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Mathematicians are not the people who find Maths easy; they are the people who enjoy how mystifying, puzzling and hard it is. Are you a mathematician?

Comment recorded on the 24 May 'Starter of the Day' page by Ruth Seward, Hagley Park Sports College:

"Find the starters wonderful; students enjoy them and often want to use the idea generated by the starter in other parts of the lesson. Keep up the good work"

Comment recorded on the 17 November 'Starter of the Day' page by Amy Thay, Coventry:

"Thank you so much for your wonderful site. I have so much material to use in class and inspire me to try something a little different more often. I am going to show my maths department your website and encourage them to use it too. How lovely that you have compiled such a great resource to help teachers and pupils.
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##### Featured Activity

This spectacular magic trick never fails to amaze people of all ages assuming they can add and subtract very simple numbers. The mathematics comes in finding out how the trick works.

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## Go Maths

Learning and understanding Mathematics, at every level, requires learner engagement. Mathematics is not a spectator sport. Sometimes traditional teaching fails to actively involve students. One way to address the problem is through the use of interactive activities and this web site provides many of those. The Go Maths page is an alphabetical list of free activities designed for students in Secondary/High school.

## Maths Map

Are you looking for something specific? An exercise to supplement the topic you are studying at school at the moment perhaps. Navigate using our Maths Map to find exercises, puzzles and Maths lesson starters grouped by topic.

## Teachers

If you found this activity useful don't forget to record it in your scheme of work or learning management system. The short URL, ready to be copied and pasted, is as follows:

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## Description of Levels

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Factorising - Factorise algebraic expressions in this structured online self marking exercise.

Level 1 - A quadratic equation presented in a factorised form.

Level 2 - Two terms where the unknown is a factor of both. The roots are integers.

Level 3 - Three terms where the squared term has a coefficient of one. The roots are integers.

Level 4 - Three terms where the squared term has a coefficient other than one and the expression factorises.

Level 5 - Quadratic equations that factorise after being rearranged.

Level 6 - The difference between two squares.

Level 7 - Three terms and the roots are not necessarily integers.

Level 8 - Mixed questions on solving quadratic equations

Exam Style Questions - A collection of problems in the style of GCSE or IB/A-level exam paper questions (worked solutions are available for Transum subscribers).

More Algebra including lesson Starters, visual aids, investigations and self-marking exercises.

Answers to this exercise are available lower down this page when you are logged in to your Transum account. If you don’t yet have a Transum subscription one can be very quickly set up if you are a teacher, tutor or parent.

## Curriculum Reference

See the National Curriculum page for links to related online activities and resources.

## Help

### Example Level 1

$$(x − 8)(4x − 5) = 0$$

Here there are two sets of brackets which multiply together to give zero. It is therefore true that the contents of at least one of the sets of brackets must be zero.

So either $$x-8=0$$ which means $$x=8$$

Or $$4x-5=0$$ which means $$4x=5$$

Therefore $$x = 5 \div 4 = 1.25$$

So the two answers are 8 and 1.25

Don't wait until you have finished the exercise before you click on the 'Check' button. Click it often as you work through the questions to see if you are answering them correctly. You can double-click the 'Check' button to make it float at the bottom of your screen.

Answers to this exercise are available lower down this page when you are logged in to your Transum account. If you don’t yet have a Transum subscription one can be very quickly set up if you are a teacher, tutor or parent.

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Question: EBImage -- local machine VS cluster -- could not find function "slot"
0
5 days ago by
Mathieu Gendarme10 wrote:
Dear all,

I am currently trying to run a script for image segmentation using EBImage. The script I used was tested on two different platform, either a local workstation running Ubuntu or on a cluster running CentOS version 7.

Everything runs fine on the workstation but on the cluster I keep getting this error:

Error in slot(nucMask, ".Data") : could not find function "slot"
Calls: ImageProcessing -> [ -> [ -> eval.parent -> eval -> eval
Execution halted

As far as I can assess all the dependencies for this script are installed and I can load both libraries required for the job.

Does anyone know how to fix this problem?

Cheers,

Mathieu

Here is the code I used:

https://www.dropbox.com/s/bd4fo0p6q4yzatk/IfTestCluster_Bioconductor.R?dl=0

written 5 days ago by Mathieu Gendarme10

How exactly do you run your script on the cluster, do you call it in a different way than locally?

The script is called from a .sh script that can be found here:

https://www.dropbox.com/s/x1477r7xvrgm0r3/IfTestCluster_Bioconductor.sh?dl=0