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# Even-odd turn game with two integers in C++

C++Server Side ProgrammingProgramming

In this problem, we are given three integer values A , B and T. Our task is to create a program to play Even-odd turn game with two integers.

The two integer value are :

T, that denotes the number of turns in the game.
A denotes the value for player1
B denotes the value for player2

If the value of T is odd, the value of A is multiplied by 2.
If the value of T is even, the value of B is multiplied by 2.
We need to find and return value of max(A, B) / min(A, B) at the end.

## Let’s take an example to understand the problem,

Input: A = 3, B = 4, T = 3

Output: 1

Explanation:

1st Turn : T is odd, A is multiplied by 2, A = 6.
2nd Turn: T is even, B is multiplied by 2, B = 8.
3rd Turn: T is odd, A is multiplied by 2, A = 12.

A = 12                  B = 4

max(A, B) = max(12, 4) = 12
min(A, B) = min(12, 4) = 4
max(A, B) / min(A, B) = 12/ 8 = 1

Solution Approach:

A simple solution to the problem would be calculating the value of A and B after T turns and then return the value of max(A, B) / min(A, B). This is an effective solution by taking T iterations.

But their can be a more effective solution based on the fact that for even value of T, the value of new A is N*A and the value of new B is N*B.

This make the value of max(A, B)  / min(A, B) a  constant which is equal to
max(A, B) / min(A, B).

If the value of T is odd, the value of A will be 2*N*A and the value of B is N*B.

This makes the value of max(A, B) / min(A, B) a constant which is equal to max(2A, B) / min(2A, B).

The result of the problem max(A, B) / min(A, B) =

max(A, B) / min(A, B), if T is even
max(A, B) / min(A, B), if T is odd

## Example

Live Demo

#include <iostream>
using namespace std;

int EvenOddGame(int A, int B, int T) {

if ( T%2 == 0)
return (max(A, B) / min(A, B));
else
return (max(2*A, B) / min(2*A, B));
return -1;
}

int main() {

int A = 3, B = 2, T = 3;
cout<<"The return value of even odd game is "<<EvenOddGame(A, B, T);

}

## Output −

The return value of even odd game is 3

Published on 22-Jan-2021 13:16:29

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# A supertrain (proper length 100 $\mathrm{m} )$ travels at a speed of 0.950 $\mathrm{c}$ as it passes through a tunnel (proper length 50.0 $\mathrm{m}$ ). As seen by a trackside observer, is the train ever completely within the tunnel? If so, with how much space to spare?

## 18.775$m$

Gravitation

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in this problem, the observer measures for permanent off journal, which is good to 50.0 imitation but measures Shane contracted to length. You lent off L which is equal to in or into one minus. We square do it by Isis where, as we were given the value of this code, would we do it by sea so well secured when we get and notice about 100 m and if you do it by C is equal toe 0.950 When we saw this, we get the contracted length of train is equal to 31.2 m. So the trans shorter than the tunnel by 50 0.0 meter, minus 31.2 m, which is equal to 18.8 m. She's the answer of the question and this length is actually less than the length afternoon. So the train is completely within the eternal

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Article Text

Single-cell RNA sequencing for the study of lupus nephritis
1. Evan Der1,
2. Hemant Suryawanshi2,
4. Thomas Tuschl2 and
5. Chaim Putterman1
1. 1Division of Rheumatology, Albert Einstein College of Medicine, Bronx, New York, USA
2. 2Laboratory for RNA Molecular Biology, Rockefeller University, New York, USA
3. 3Division of Rheumatology, Department of Medicine, New York University School of Medicine, New York, USA
1. Correspondence to Dr Chaim Putterman; chaim.putterman{at}einstein.yu.edu

## Abstract

Single-cell RNA sequencing (scRNA-seq) has recently undergone rapid advances in the development of this technology, leading to high throughput and accelerating discovery in many biological systems and diseases. The single-cell resolution of the technique allows for the investigation of heterogeneity in cell populations, and the pinpointing of pathological populations contributing to disease. Here we review the development of scRNA-seq technology and the analysis that has evolved with the ever-increasing throughput. Finally, we highlight recent applications of scRNA-seq to understand the molecular pathogenesis of lupus and lupus nephritis.

• lupus nephritis
• systemic lupus erythematosus
• inflammation

This is an Open Access article distributed in accordance with the Creative Commons Attribution Non Commercial (CC BY-NC 4.0) license, which permits others to distribute, remix, adapt, build upon this work non-commercially, and license their derivative works on different terms, provided the original work is properly cited and the use is non-commercial. See: http://creativecommons.org/licenses/by-nc/4.0

View Full Text

## Introduction

Kidney involvement in patients with systemic lupus erythematosus (SLE), or lupus nephritis (LN), can affect up to 50% or more of all lupus patients and remains an important source of morbidity and mortality. Although the exact mechanism of LN progression is still a matter of ongoing research and debate, there are several distinctive features of the disease including immune complex deposition on the glomerular basement membrane, which can result in inflammation and ultimately reduced glomerular function.1 Additionally, hyperproliferation of the resident mesangial and endothelial cells, and fibrosis affecting the renal interstitial and glomerular compartments are often observed as the disease progresses.2 3 Examination of renal tissue obtained by a percutaneous biopsy reveals that glomerular involvement can be highly variable even within the same patient. Moreover, patients with a similar histological classification can have variable responses to treatment, even when similar protocols are employed. Thus, the examination of kidney biopsy tissue by light and electron microscopy alone do not provide sufficiently accurate prognostic information, and additional modes for assessing disease progression must be developed to optimise patient care. Lupus renal biopsies demonstrate notable histological differences between patients which are reflected in the classification scheme in widespread use; however, the mechanisms underlying these distinct histopathological classes are not well understood, and at the molecular level, they may even display more heterogeneity than the proliferative and membranous patterns currently recognised. Finally, studies of LN and kidney diseases, in general, have traditionally relied on visualisation-based approaches such as microscopy and flow cytometry. These approaches do not provide sufficient data regarding gene expression changes within specific cell types. Such an understanding may further elucidate the communication between cells and the pathological processes contributing to heterogeneity in ultimate tissue injury.

More recently, approaches such as transcriptome profiling with microarrays and RNA sequencing (RNA-seq) have been applied in LN to RNA isolated from whole or laser-capture micro-dissected renal tissue. For instance, Fu et al used RNA-seq to elucidate the molecular basis for the efficacy of combination therapy in LN, finding that combination therapy with tacrolimus and mycophenolate mofetil as compared with either drug given alone suppressed the IL-6/STAT3 pathway and inhibited Toll-like receptor 7 expression.4 ‘Bulk’ RNA sequencing of tissues yields information regarding the average expression of all cell types contributing to the isolated RNA. Therefore, even in cases where glomeruli alone are sequenced, expression changes may come from any of the cell types found within the glomerulus and cannot be unambiguously assigned to a specific cell type. This can potentially cause changes in interesting genes to be overlooked within the average especially if they are expressed in a rare cell type which may be only present at the level of a few percents of a tissue. Although cytometric sorting of cell populations before sequencing can greatly increase the cell type specificity of RNA-seq analysis, heterogeneity within a cell type such as proliferating versus resting cells still may remain unrecognised.

Single-cell RNA-seq (scRNA-seq) addresses these technical challenges by providing transcriptomic profiling at a single-cell resolution. This methodology allows for the characterisation of cells represented in tissue and independent from each other without prior knowledge of lineage markers.5 Furthermore, scRNA-seq facilitates the identification of cell-state-specific expression patterns and offers unique insights into cell differentiation and cell–cell ligand–receptor interactions. Advances in microfluidic technology have lowered the cost per cell and dramatically increased the throughput, propelling the technology into a new age. In this paper, we review the promise of scRNA-seq and its potential application to different diseases involving the kidney.

### scRNA-seq Protocols

scRNA-seq is fundamentally similar to traditional RNA-seq in that it involves reverse transcription and preparation of a cDNA library followed by DNA sequencing. The most important difference is that scRNA-seq yields many libraries carrying cell-specific barcodes instead of one single library from the RNA pool of many cells. In bulk RNA-seq, small cell populations contribute little when averaged with larger cell populations, but in scRNA-seq, they contribute unique profiles at a rare frequency (figure 1). This is accomplished in different ways depending on the technique and can be broadly categorised into two distinct approaches: (a) plate/tube-based and (b) microfluidic-based strategies. These two categories share common workflows starting with the isolation of a single-cell suspension from tissue or sample, physical capture of single cells in distinct chambers, lysis, reverse transcription, amplification, library preparation and next-generation sequencing. scRNA-seq methods differ in the way single cells are captured and physically separated and in the chemistry used to amplify and create libraries for sequencing (figure 2). Regardless, the amplification step can introduce biases due to the kinetics of cDNA amplification leading to artefactual over-representation of certain mRNAs. This has been dealt with in many scRNA-seq protocols by the integration of unique molecular identifiers (UMIs) which tag sequence reads originating from an mRNA molecule. UMIs provide a digital expression approximating actual count of mRNAs sequenced rather than relative abundance which is typical of traditional RNA-seq, and in this way remove PCR bias. scRNA-seq originated as a tube-based or plate-based methodology and evolved as microfluidic technology advanced, each evolution increasing the throughput from single-digit numbers of profiles per experiment to thousands per preparation.

Figure 1

Bulk sequencing versus scRNA-seq. In bulk sequencing, a pool of extracted RNA from a population of cells is sequenced, providing expression data representing the average expression of a particular gene across all cells. In contrast, scRNA-seq retains the originating cell-specific transcript information. If many cells of the same cell type were identified, the averaging of sequence reads across all cells yields cell-type-specific expression information similar to bulk RNA-seq profiles. Hypothetical outputs for a gene in bulk sequencing and scRNA-seq are shown. scRNA, single-cell RNA sequencing; tSNE, t-Distributed Stochastic Neighbour Embedding.

Figure 2

ScRNA-seq workflows. The basic workflow of scRNA-seq follows similar steps regardless of the platform. Tissue is disaggregated into single-cell suspensions and loaded onto the scRNA-seq platform of choice (ie, Dropseq or Fluidigm C1). Lysis, reverse transcription, PCR amplification and sequencing are followed by downstream analyses. scRNA, single-cell RNA sequencing.

Plate-based and tube-based methods include the original method of scRNA-seq by Tang et al where individual cells were manually transferred into PCR tubes for reverse transcription, amplification and library preparation.6 Single-cell tagged reverse transcription (STRT-Seq)7 and switching mechanism at 5’ end of RNA template (SMART-Seq)8 both increased the scale of scRNA-seq by transitioning to a 96-well plate format where cell barcodes were introduced during the library preparation step. This allowed for the pooling and sequencing of 96 single-cell libraries in a single sequencing run, decreasing the cost of sequencing and increasing the throughput. Cell expression by linear amplification and sequencing2 (CEL-Seq2)9 improved further on this method by introducing UMIs into the reverse transcription allowing for the reduction in noise due to non-linear PCR amplification. This process furthermore relies on 3’ end counting and thus does not generate full-length sequences of cDNA, which precludes the examination of splice variation and isoform. When paired with an automated cell capture method such as fluorescence activated cell sorting (FACS), or the Fluidigm C1 integrated fluidic chip,10 the automation resulted in an order of magnitude increase in throughput. The Fluidigm C1 system has the added benefit of reducing the reaction volumes from μL to nL volumes for reverse transcription and improves on the efficiency of some enzymatic steps during cDNA library preparation. In addition, this system allows imaging of individual cells before processing. Fluidigm also released a high throughput platform with an integrated fluidic chip capable of capturing 800 cells per experiment instead of 96 per experiment in the previous generation, further increasing the throughput capacity in scRNA-seq. The cost per cell, however, remains relatively high with this level of throughput, approximately US$3–5 per cell. ### Droplet-based scRNA-seq Droplet-based approaches utilise microfluidic technology to capture single cells in buffer-oil emulsion droplets for cDNA capture and/or reverse transcription with droplet volumes in the nL range. This technology has the largest throughput potential, limited only by sample material and cost. The best technologies conduct reverse transcription and barcoding within the droplet. The first microfluidic-based approaches were DropSeq11 and InDrops,12 which were developed and published at the same time. Both methods rely on a microfluidic chip to control the flow of oil, cell suspension and barcoded beads suspended in lysis buffer and reaction reagents, for optimal droplet formation. Millions of droplets form at the intersection of the three inlets with the aim of simultaneously capturing one cell and one bead per droplet. One of the critical components of these methods are the barcoded beads, which in the case of DropSeq are microparticle beads covered with oligonucleotides containing a PCR handle, a unique cell barcode (UMI) and a 3′ terminal polyT segment typically 30 nucleotides in length. The polyT segment is for capturing polyA mRNA and priming of reverse transcription, and although the polyA tail length can vary, a 30-nucleotide oligo-dT is sufficient for the capture of mature mRNAs. During reverse transcription, the cell barcode, UMI and PCR handle are incorporated into each cDNA molecule. This allows for the pooling of all cDNAs during subsequent PCR steps as the cell identity and UMIs are specific for each cDNA (figure 3). Although millions of droplets can be formed in this manner, the beads and cell suspension flow rates need to be adjusted to minimise doublet cell capture, yet maximise the frequency of a droplet containing both a cell and a barcoded bead. The capture rates for DropSeq were suboptimal and near 5%–10% of total input cells. InDrops uses hydrogel beads packed with oligonucleotides. These beads are deformable and can be passed through narrower channels in comparison with DropSeq solid beads, which allows for improved fluidic control and increased cell capture rates of 60%–90%. Both methods significantly reduced the cost per cell over the plate-based scRNA-seq methods to US$0.60–0.90, though the total cost per experiment overall remains similar.

Figure 3

Droplet-based scRNA-seq integration of cell barcodes and UMIs. Droplet-based scRNA-seq platforms use barcoded beads to retain information both on the cell of origin and mRNA sequence. This is accomplished by use of beads conjugated to oligonucleotides-containing sequences for the PCR priming, a cell barcode unique for each bead and a UMI randomised sequence distinguishing oligonucleotides per bead. The end of the oligo is a polyT tail which binds to polyA tails of mature mRNAs. This allows reverse transcription of the mRNA into a cDNA molecule which now contains the PCR handle, cell barcode and UMI. in this way, during PCR amplification, the information regarding input cell of origin and mRNA of origin is retained. When the library is prepared and sequenced, both ends are sequenced. The first read will sequence the cell barcode and UMI, whereas the second read sequences the cDNA. Together these reads provide the gene encoding the mRNA, the cell it was expressed in, and further, by summing UMIs, the total number of mRNAs for the specific gene in that cell. scRNA, single-cell RNA sequencing; UMI, unique molecular identifiers.

Several commercial adaptations of microfluidic scRNA-seq approaches have emerged after the DropSeq and InDrops publications. The most notable two are the 10X Genomics Chromium platform and the Illumina/Bio-Rad system. Both are more expensive per cell than DropSeq or InDrops custom-built devices, but offer more standardisation and convenience, analytical software and technical support. 10X Genomics has recently introduced an approach to also capture the 5′ ends of mRNAs. This enables T-cell receptor and B-cell receptor sequencing in tandem with mRNA 3′ end sequencing, in order to evaluate the clonality of B and T cells.

### Protein integration

One of the major limitations of scRNA-seq is that not all mRNAs are captured by this approach and only a small percentage are sampled so that statistical methods need to be employed for assigning cell-type identity. Furthermore, the sparse sampling of transcripts per cell is prohibitive to estimating RNA abundance per cell. However, lineage assignment and combination of reads from the same cell types ultimately leads to cell-type-specific expression profiles, as obtained from ‘bulk’ RNA sequencing from purified cell populations. Sparse sampling is a particular problem when assigning cell states of small cell types with little cytoplasm and lower mRNA content, especially leukocytes. This has prompted efforts of combining scRNA-seq with simultaneous antibody-based protein detection referred to as RNA Expression And Protein sequencing assay (REAP-seq) 13 and Cellular Indexing of Transcriptomes and Epitopes by sequencing (CITE-seq).14 Similar to flow cytometry, but instead of labelling antibodies with a fluorophore or heavy metal ion tags, they are conjugated to polyA-tail-containing oligodeoxynucleotides to be sequenced simultaneously with the single-cell transcriptome. The initial proof of concept experiment labelled with 16 distinct antibodies. Although promising, these techniques have not yet been applied to kidney tissue, and it is likely that further careful optimisation of each antibody will be necessary to ensure that the relevant epitopes are not cleaved during the tissue disaggregation. The potential for surface protein expression to be altered during preparation for scRNA-seq has also not yet been explored in kidney cells.

### Data analysis

Nearly every scRNA-seq platform relies on Illumina sequencing to provide sequence read files in FASTA or FASTQ format for downstream analyses. However, due to the distinct barcode, UMI and sequencing adapter usage, each methodology also relies on custom bioinformatic pipelines, whereas commercial platforms may additionally distribute analysis software with their platforms. Common to these pipelines, FASTQ sequence read files are trimmed from adapters and need to pass filters controlling for sequencing quality. Sequence reads are then aligned against the genome in order to assign the reads to genes. Platforms using UMIs generate a digital expression matrix based on a number of transcripts captured per single cell. Platforms that do not incorporate UMIs generate a matrix composed of relative transcript abundances.

Following the computation of the expression matrix, many different kinds of analyses may be performed which is also an area of active new development. The most prevalent analysis is dimensionality reduction analysis such as principal component analysis and t-Distributed Stochastic Neighbour Embedding11 to visualise clusters of similar cells identified by unsupervised graph-based population identification algorithms (figure 4A). These tools facilitate the detection of novel cell types and subpopulations of cells representing different cell states, such as resting versus proliferating. Pathways of differentiation and cell lineage positioning can be studied by ‘pseudotime’15 analysis integrating computational machine learning approaches (figure 4B).

Figure 4

Representative plots of downstream scRNA-seq analysis. (A) An illustrative tSNE plot, where cells form clusters based on similarities and differences in gene expression. Different colours designate different clusters which differentiate between cell types and cell states. (B) An illustrative pseudotime plot, the line indicating a continuum of hypothetical differentiation from least differentiated (red colour) to several branches of distinct differentiated cell types. scRNA, single-cell RNA sequencing; tSNE, t-Distributed Stochastic Neighbour Embedding.

There are several recognised limitations of scRNA-seq. First, certain cell types sensitive to tissue dissociation may be lost or under-represented and the relative cell counts may not be representative of the original tissue. Second, although gene expression analysis by scRNA-seq provides a very good representation of the expression pattern found by conventional bulk sequencing approaches, application of scRNA-seq shares a limitation present also in the standard method of biopsy analysis, and that is, both methods analyse only a relatively small piece of the whole tissue which may not be sufficiently representative of the entire organ. Third, due to incomplete sampling of cellular gene expression, statistical approaches are required for cell type identification and clustering. Multiple imputation-based software packages have been established to computationally address this issue.16 17 Last, spatial information is not preserved as tissue needs to be dissociated for preparing single-cell suspension. Considering the rapid development of scRNA-seq methods and instruments, specific software platforms were developed for integrative analysis of data generated on diverse platforms.

### Clinical implications

This emerging technology is already being applied to gain insight into the pathogenesis and heterogeneity of LN and other organ manifestations of SLE. Transcriptomic evaluation at the single-cell level is poised to identify rare cell populations and evaluate the communication between renal cell types and inflammatory leukocytes and fibroblasts. Although other fields such as cancer biology and neurosciences have already begun to leverage this technology, it has yet to be used extensively within the context of SLE and LN. Several groups, however, have begun characterising healthy and diseased kidney in murine models and humans with LN. These studies provide a single-cell atlases of the kidney, for instance, Brunskill et al used the C1/SMART-Seq platform to characterise renal organogenesis in developing mice and provide a roadmap of renal development.18 More recently, Park et al used the 10X Genomics platform to sequence 57 979 healthy adult mouse kidney cells.19 This undertaking provided a detailed resolution of the cell populations of the mouse kidney and identified novel transition states between cell types and potentially novel populations of collecting duct cells. Specifically evaluating targets reported in GWAS studies of various renal diseases including tubular acidosis and nephrotic syndrome, the authors identified the candidate genes expressed in specific renal cell types indicating a possible cell-type-specific contribution of each disease. Other studies of mouse kidney by scRNA-seq have focused on specific sorted cell types including mesangial cells and collecting duct cells.

In humans, Sivakamasundari et al performed a similar analysis using the 10X Genomics platform, sequencing 22 469 healthy human kidney cells.20 Similar to Park et al, novel cell populations as well as new markers for already established cell types were identified. Analysis of the specific expression of known LN-associated genes, such as NOTCH4, indicated that their expression under healthy conditions is high in endothelial cells, specifically ascending and descending vasa recta. Our group has recently characterised renal and skin biopsies from patients with LN and live organ donors using the Fluidigm C1/SMART-Seq platform.21 Single-cell transcriptomic analysis revealed an interferon response signature that correlated with clinical parameters, including histological features such as glomerular immunoglobulin G deposition and chronicity. Further, a high interferon response signature was associated with poor response to conventional treatment. This finding was validated in a separate cohort using the high throughput Fluidigm platform and we have further identified a fibrotic signature that could predict response to treatment at 6 months postbiopsy.22 The higher throughput and larger patient cohort in this latter study allowed for biopsy class-specific comparisons which revealed several pathways that may contribute to the histological differences in the disease. Another group has used a plate-based CEL-SEQ2 approach on the leukocytes sorted from the kidneys of LN patients and have reported the immune landscape of the kidney. They were able to identify several subsets of immune cells including cells that appear to be T follicular helper regulatory cells.23 This work sets the stage for new discoveries, including an increased understanding of LN pathophysiology that will lead to novel and more targeted therapies. Further, adding precision diagnostics and prognostics will help guide clinical treatment decisions. For example, preliminary data using droplet-based approaches have indicated relatively high variability in terms of the number and type of infiltrating immune cells within the interstitial and glomerular compartments between patients, even in those identified with the same histopathology class (data not shown). If this result is confirmed, such information could drive the choice of B-cell depleting approaches in patients with high numbers of kidney-infiltrating B cells. Similarly, early signs of extracellular matrix expression in tubular cells, even in the absence of other evidence for fibrosis and chronicity, may suggest antifibrotic approaches. Another important clinical application would be a move towards therapy intensification, or at least shorter follow-up intervals, in those patients whose gene expression patterns display negative prognostic indicators. scRNA-seq lays the foundation to usher in an era of precision medicine where it is anticipated that therapies based on renal transcriptomics will increase response rates and guide patient-specific treatment decisions.

## Acknowledgments

We would like to thank the Accelerating Medicines Partnership (AMP) network and investigators for their intellectual contribution and work towards advancing single-cell RNAseq in lupus nephritis.

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View Abstract

## Footnotes

• Contributors ED, HS, TT and CP prepared and wrote the manuscript.

• Funding Funding was provided through grants from the National Institutes of Health (UH2-AR067676, UH2-AR067677, UH2-AR067679, UH2-AR067681, UH2-AR067685, UH2-AR067688, UH2-AR067689, UH2-AR067690, UH2-AR067691, UH2-AR067694, and UM2-AR067678).

• Competing interests None declared.

• Patient consent for publication Not required.

• Provenance and peer review Commissioned; externally peer reviewed.

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# 基于SVM的雷达故障预诊断技术研究The Pre-diagnosis Technology Research on SVM-based Radar Fault

Abstract: This paper will Support Vector Machine (SVM) bring to the pre-diagnosis of radar fault, implementing the advance diagnosis of the latent radar fault. To slove the problems best, such as the process of radar fault diagnosis complexity and long-time diagnosis. This is due to SVM can solve the small sample, nonlinear classification problem, which is the characteristics of the latent radar fault. Finally, the paper through examples fully illustrates that the algorithm has the capacity of excluding the radar latent fault.

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### Cryptographic Agents: Towards a Unified Theory of Computing on Encrypted Data

Shashank Agrawal, Shweta Agrawal, and Manoj Prabhakaran

##### Abstract

We provide a new framework of cryptographic agents that unifies various modern cryptographic objects'' --- identity-based encryption, fully-homomorphic encryption, functional encryption, and various forms of obfuscation -- similar to how the Universal Composition framework unifies various multi-party computation tasks like commitment, coin-tossing and zero-knowledge proofs. These cryptographic objects can all be cleanly modeled as schemata'' in our framework. Highlights of our framework include the following: - We use a new indistinguishability preserving' (INDPRE) definition of security that interpolates indistinguishability and simulation style definitions, which (often) sidesteps the known impossibilities for the latter. INDPRE-security is parameterized by the choice of the test'' family, such that by choosing different test families, one can obtain different levels of security for the same primitive (including various standard definitions in the literature). - We present a notion of reduction' from one schema to another and a powerful composition theorem' with respect to INDPRE security. We show that obfuscation is a complete'' schema under this notion, under standard cryptographic assumptions. We also provide a stricter notion of reduction that composes even when security is only with respect to certain restricted test families of importance. - Last but not the least, our framework can be used to model abstractions like the generic group model and the random oracle model, letting one translate a general class of constructions in these heuristic models to constructions based on standard model assumptions'. We also illustrate how our framework can be applied to specific primitives like obfuscation and functional encryption. We relate our definitions to existing definitions and also give new constructions and reductions between different primitives.

Note: Reorganization and minor corrections.

Available format(s)
Category
Foundations
Publication info
A minor revision of an IACR publication in EUROCRYPT 2015
Contact author(s)
sagrawl2 @ illinois edu
shweta a @ gmail com
mmp @ illinois edu
History
2015-05-01: last of 2 revisions
See all versions
Short URL
https://ia.cr/2014/480

CC BY

BibTeX

@misc{cryptoeprint:2014/480,
author = {Shashank Agrawal and Shweta Agrawal and Manoj Prabhakaran},
title = {Cryptographic Agents: Towards a Unified Theory of Computing on Encrypted Data},
howpublished = {Cryptology ePrint Archive, Paper 2014/480},
year = {2014},
note = {\url{https://eprint.iacr.org/2014/480}},
url = {https://eprint.iacr.org/2014/480}
}

Note: In order to protect the privacy of readers, eprint.iacr.org does not use cookies or embedded third party content.


text:
GATE ECE
General Aptitude
Numerical Ability
Previous Years Questions

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New sequential Lagrange multiplier conditions characterizing optimality without constraint qualification for convex programs. (English) Zbl 1046.90059
Summary: In this paper a new sequential Lagrange multiplier condition characterizing optimality without a constraint qualification for an abstract nonsmooth convex program is presented in terms of the subdifferentials and the $ϵ$-subdifferentials. A sequential condition involving only the subdifferentials, but at nearby points to the minimizer for constraints, is also derived. For a smooth convex program, the sequential condition yields a limiting Kuhn-Tucker condition at nearby points without a constraint qualification. It is shown how the sequential conditions are related to the standard Lagrange multiplier condition. Applications to semidefinite programs, semi-infinite programs, and semiconvex programs are given. Several numerical examples are discussed to illustrate the significance of the sequential conditions.
MSC:
 90C25 Convex programming 52A41 Convex functions and convex programs (convex geometry) 26E15 Calculus of real functions on infinite-dimensional spaces


text:
## CW Approximation

A weak homotopy equivalence is a map $f:X\to Y$ that induces isomorphisms $\pi_n(X,x_0)\to\pi_n(Y,f(x_o))$ for all $n\geq 0$ and all choices of basepoint $x_0$.

In other words, Whitehead’s theorem says that a weak homotopy equivalence between CW complexes is a homotopy equivalence. Just to recap, a map $f:X\to Y$ is said to be a homotopy equivalence if there exists a map $g:Y\to X$ such that $fg\cong id_Y$ and $gf\cong id_X$. The spaces $X$ and $Y$ are called homotopy equivalent.

It turns out that for any space $X$ there exists a CW complex $Z$ and a weak homotopy equivalence $f:Z\to X$. This map $f:Z\to X$ is called a CW approximation to $X$.

## Excision for Homotopy Groups

According to Hatcher (Chapter 4.2), the main difficulty of computing homotopy groups (versus homology groups) is the failure of the excision property. However, under certain conditions, excision does hold for homotopy groups:

Theorem (4.23): Let $X$ be a CW complex decomposed as the union of subcomplexes $A$ and $B$ with nonempty connected intersection $C=A\cap B$. If $(A,C)$ is m-connected and $(B,C)$ is n-connected, $m,n\geq 0$, then the map $\pi_i(A,C)\to\pi_i(X,B)$ induced by inclusion is an isomorphism for $i and a surjection for $i=m+n$.

## Miscellaneous Definitions

Suspension: Let $X$ be a space. The suspension $SX$ is the quotient of $X\times I$ obtained by collapsing $X\times\{0\}$ to one point and $X\times\{1\}$ to another point.

The definition of suspension is similar to that of the cone in the following way. The cone $CX$ is the union of all line segments joining points of $X$ to one external vertex. The suspension $SX$ is the union of all line segments joining points of $X$ to two external vertices.

The classical example is $X=S^n$, when $SX=S^{n+1}$ with the two “suspension points” at the north and south poles of $S^{n+1}$, the points $(0,\dots,0,\pm 1)$.

Here are some graphical sketches of the case where $X$ is the 0-sphere and the 1 sphere respectively.

## Cellular Approximation for Pairs

(This is Example 4.11 in Hatcher’s book).

Cellular Approximation for Pairs: Every map $f:(X,A)\to (Y,B)$ of CW pairs can be deformed through maps $(X,A)\to (Y,B)$ to a cellular map $g:(X,A)\to (Y,B)$.

What “map of CW pairs” mean, is that $f$ is a map from $X$ to $Y$, and the image of $A\subseteq X$ under $f$ is contained in $B$CW pair $(X,A)$ means that $X$ is a cell complex, and $A$ is a subcomplex.

First, we use the ordinary Cellular Approximation Theorem to deform the restriction $f:A\to B$ to be cellular. We then use the Homotopy Extension Property to extend this to a homotopy of $f$ on all of $X$. Then, use Cellular Approximation Theorem again to deform the resulting map to be cellular staying stationary on $A$.

We use this to prove a corollary: A CW pair $(X,A)$ is n-connected if all the cells in $X-A$ have dimension greater than $n$. In particular the pair $(X,X^n)$ is n-connected, hence the inclusion $X^n\hookrightarrow X$ induces isomorphisms on $\pi_i$ for $i and a surjection on $\pi_n$.

First we note that being n-connected means that the space is non-empty, path-connected, and the first n homotopy groups are trivial, i.e. $\pi_i(X)\cong 0$ for $1\leq i\leq n$.

Proof: First, we apply cellular approximation to maps $(D^i,\partial D^i)\to (X,A)$ with $i\leq n$, thus the map is homotopic to a cellular map of pairs $g$. Since all the cells in $X-A$ have dimension greater than $n$, the n-skeleton of $X$ must be inside $A$. Therefore $g$ is homotopic to a map whose image is in $A$, and thus it is 0 in the relative homotopy group $\pi_i(X,A)$. This proves that the CW pair $(X,A)$ is n-connected. Note that 0-connected means path-connected.

Consider the long exact sequence of the pair $(X,X^n)$:

$\dots\to\pi_n(X^n,x_0)\xrightarrow{i_*}\pi_n(X,x_0)\xrightarrow{j_*}\pi_n(X,X^n,x_0)\xrightarrow{\partial}\pi_{n-1}(X^n,x_0)\to\dots\to\pi_0(X,x_0)$

Since it is an exact sequence, the image of any map equals the kernel of the next. Thus, $\text{Im}(i_*)=\ker j_*=\pi_n(X,x_0)$ (since $\pi_n(X,X^n,x_0)=0$). Thus $i_*$ is surjective. Since $\pi_n(X,X^n,x_0)=0$, the later terms in the long exact sequence are also 0, thus, the inclusion $X^n\hookrightarrow X$ induces isomorphisms on $\pi_i$ for $i, since the first n homotopy groups all vanish.

## Cellular Approximation Theorem and Homotopy Groups of Spheres

First we will state another theorem, Whitehead’s Theorem: If a map $f:X\to Y$ between connected CW complexes induces isomorphisms $f_*:\pi_n(X)\to\pi_n(Y)$ for all $n$, then $f$ is a homotopy equivalence. If $f$ is the inclusion of a subcomplex $X\to Y$, we have an even stronger conclusion: $X$ is a deformation retract of $Y$.

The main theorem discussed in this post is the Cellular Approximation Theorem: Every map $f:X\to Y$ of CW complexes is homotopic to a cellular map. If $f$ is already cellular on a subcomplex $A\subset X$, the homotopy may be taken to be stationary on $A$. This theorem can be viewed as the CW complex analogue of the Simplicial Approximation Theorem.

Corollary: If $n, then $\pi_n(S^k)=0$.

Proof: Consider $S^n$ and $S^k$ with their canonical CW-structure, with one 0-cell each, and with one n-cell for $S^n$ and one k-cell for $S^k$. Let $[f]\in\pi_n(S^k)$, where $f:S^n\to S^k$ is a base-point preserving map. By the Cellular Approximation Theorem, $f$ is homotopic to a cellular map $g$, where cells map to cells of same or lower dimension.

Since $n, the n-cell $S^n$ can only map to the 0-cell in $S^k$. The 0-cell in $S^n$ (the basepoint) is also mapped to the 0-cell in $S^k$. Thus $g$ is the constant map, hence $\pi_n(S^k)=0$.

## Covering space projection induces isomorphisms

Proposition 4.1 (from Hatcher): A covering space projection $p:(\tilde{X},\tilde{x}_0)\to (X,x_0)$ induces isomorphisms $p_*:\pi_n(\tilde{X},\tilde{x}_0)\to\pi_n(X,x_0)$ for all $n\geq 2$.

We will elaborate more on this proposition in this blog post. Basically, we will need to show that $p_*$ is a homomorphism and also bijective (surjective and injective).

Homomorphism

$p_*([f]):=[pf]$

$p_*([f]+[g])=[p(f+g)]$

$p(f+g)(s_1,s_2,\dots,s_n)=\begin{cases}pf(2s_1,s_2,\dots,s_n)&s_1\in[0,\frac 12]\\ pg(2s_1-1,s_2,\dots,s_n)&s_1\in[\frac 12,1] \end{cases}$

$p_*[f]+p_*[g]=[pf]+[pg]$

$(pf+pg)(s_1,s_2,\dots,s_n)= \begin{cases}pf(2s_1,s_2,\dots,s_n)&s_1\in[0,\frac 12]\\ pg(2s_1-1,s_2,\dots,s_n)&s_1\in[\frac 12,1] \end{cases}$, which we can see is the same.

Thus, $p_*$ is a homomorphism.

Surjective

For surjectivity, we need to use a certain Proposition 1.33: Suppose given a covering space $p:(\tilde{X},\tilde{x}_0)\to (X,x_0)$ and a map $f:(Y,y_0)\to (X,x_0)$ with $Y$ path-connected and locally path-connected. Then a lift $\tilde{f}:(Y,y_0)\to (\tilde{X},\tilde{x}_0)$ of $f$ exists iff $f_*(\pi_1(Y,y_0))\subset p_*(\pi_1(\tilde{X},\tilde{x}_0))$.

Let $[f]\in\pi_n(X,x_0)$, where $f:(S_n,s_0)\to(X,x_0), n\geq 2$. Since $S^n$ is simply connected for $n\geq 2$, $\pi_1(S_n,s_0)=0$. Thus $f_*(\pi_1(S_n,s_0))=0\subset p_*(\pi_1(\tilde{X},\tilde{x}_0))$. By Proposition 1.33, a lift $\tilde{f}:(S_n,s_0)\to (\tilde{X},\tilde{x}_0)$ of $f$ exists, where $p\tilde{f}=f$.

i.e. we have $\boxed{p_*[\tilde{f}]=[p\tilde{f}]=[f]}$. Hence $p_*$ is surjective.

Injective

Let $[\tilde{f}_0]\in\ker p_*$, where $\tilde{f}_0:I^n\to \tilde{X}$ with a homotopy $f_t:I^n\to X$ of $f_0=p\tilde{f}_0$ to the trivial loop $f_1$.

By the covering homotopy property (homotopy lifting property), there exists a unique homotopy $\tilde{f}_t:I^n\to \tilde{X}$ of $\tilde{f}_0$ that lifts $f_t$, i.e. $p\tilde{f}_t=f_t$. There is a lifted homotopy of loops $\tilde{f}_t$ starting with $\tilde{f}_0$ and ending with a constant loop. Hence $[\tilde{f}_0]=0$ in $\pi_n(\tilde{X},\tilde{x}_0)$ and thus $p_*$ is injective.

## The Groupoid Properties of Operation * on Path-homotopy Classes (Proof)

Theorem: The operation * has the following properties:

(1) (Associativity) [f]*([g]*[h])=([f]*[g])*[h], i.e. it doesn’t matter where we place the brackets.

(2) (Right and left identities) Given $x\in X$, let $e_x$ denote the constant path $e_x: I\to X$ mpping all of I to the point x. If f is a path in X from $x_0$ to $x_1$, then $[f]*[e_{x_1}]=[f]$ and $[e_{x_0}]*[f]=[f]$.

(3) (Inverse) Given the path f in X from $x_0$ to $x_1$, let $\bar{f}$ be the path defined by $\bar{f}=f(1-s)$. $\bar{f}$ is called the reverse of f. Then, $[f]*[\bar{f}]=[e_{x_0}]$ and $[\bar{f}]*[f]=[e_{x_1}]$.

We will prove the above statements, of which (1) Associativity is actually the trickiest.

Proof:

We shall prove two elementary lemmas first. (This part is not proved in the book by Munkres).

Lemma 1: If $k: X\to Y$ is a continuous map, and if F is a path homotopy in X between the paths f and f’, then $k\circ F$ is a path homotopy in Y between the paths $k\circ f$ and $k\circ f'$.

Proof of Lemma 1: Since F is a path homotopy in X between paths f and f’, we have by definition that F(s,0)=f(s), F(s,1)=f'(s), F(0,t)=x_0, F(1,t)=x_1.

Then, k F(s,0)=kf(s), kF(s,1)=kf'(s), kF(0,t)=k(x_0), kF(1,t)=k(x_1). Since kF is continuous (composition of two continuous functions), kF is inded a path homotopy in Y between he paths kf and kf’.

Lemma 2: If $k:X\to Y$ is a continuous map and if f and g are paths in X with f(1)=g(0), then

$k\circ (f*g)=(k\circ f)*(k \circ g)$

Proof of Lemma 2:

$k(f*g)(s)=kh(s)$, where h=f*g as defined previously.

$(kf)*(kg)(s)=kh(s)$.

We will first verify property (2) on Right and Left Identities. Let $e_{x_0}$ denote the constant path in I at 0, and we let $i: I\to I$ denote the identity map, which is a path in I from 0 to 1. Then $e_0 * i$ is also a path in I from 0 to 1.

Because I is convex, there is a path homotopy G in I between i and $e_0 *i$ (Straight-line homotopy) Then $f\circ G$ is a path homotopy in X between the paths $f\circ i=f$ and $f\circ (e_0 *i)$ (Lemma 1). Furthermore by Lemma 2, $f\circ (e_0 *i) = (f \circ e_0) * (f \circ i)$ which is equivalent to $e_{x_0} *f$.

A similar argument, using the fact that if $e_1$ denotes the constant path at 1, then $i*e_i$ is path homotopic in I to the path i, shows that $[f]*[e_{x_1}]=[f]$.

To prove (3) (Inverse), we note that the reverse of i is $\bar{i}(s)=1-s$. Then $i*\bar{i}$ is a path in I beginning and ending at 0. The constant path $e_0$ is also beginning and ending at 0. Again, because I is convex, there is a path homotopy H in I between $e_0$ and $i*\bar{i}$ (straight-line homotopy). Then, using lemma 1 and 2, $f\circ H$ is path homotopy between $f\circ e_0=e_{x_0}$ and $f\circ (i*\bar{i})=(f\circ i)*(f\circ\bar{i})=f*\bar{f}$. Very similarly, we can use the fact that $\bar{i}*i$ is path homotopic  in I to $e_1$ to show that $[\bar{f}]*[f]=[e_{x_1}]$.

We will continue the proof of associativity (which is longer) in the next blog post.

## The Groupoid Properties of * on Path-homotopy Classes

This is one of the first instances where algebra starts to appear in Topology. We will continue our discussion of material found in Topology (2nd Economy Edition) by James R. Munkres.

First, we need to define the binary operation *, that will later make * satisfy properties that are very similar to axioms for a group.

Definition: If f is a path in X from $x_0$ to $x_1$, and if g is a path in X from $x_1$ to $x_2$, we define the product f*g of f and g to be he path h given by the equations

$h(s)=\begin{cases}f(2s) &\text{for }s\in [0,\frac{1}{2}], \\ g(2s-1)& \text{for }s\in[\frac{1}{2}, 1]\end{cases}$

Well-defined: The function h is well-defined, at s=1/2, f(1)=x_1, g(0)=x_1.

Continuity: h is also continuous by the pasting lemma.

h is a path in X from x_0 to x_2. We think of h as the path whose first half is the path f and whose second half is the path g.

We will verify that the product operation on paths induces a well-defined operation on path-homotopy classes, defined by the equation $[f]*[g]=[f*g]$

Let F be a path homotopy between f and f’, and let G be a path homotopy between g and g’.

i.e. we have F(s,0)=f(s), F(s,1)=f'(s)
F(0,t)=x_0, F(1,t)=x_1
G(s,0)=g(s), G(s,1)=g'(s)
G(0,t)=x_1, G(1,t)=x_2

We can define:

$H(s,t)=\begin{cases}F(2s,t) &\text{for }s\in[0,\frac{1}{2}],\\ G(2s-1,t)&\text{for }s\in[\frac{1}{2},1]\end{cases}$.

We can check that F(1,t)=x_1=G(0,t) for all t, hence the map H is well-defined. H is continuous by the pasting lemma.

Let’s check that H is the required path homotopy between f*g and f’*g’.

For s in [0,1/2],

H(s,0) = F(2s,0) =f(2s)=h(s)

H(s,1) = F(2s,1) =f'(2s)=h'(s)
h’ := f’ * g’

H(0,t) = F(0,t) = x_0

s in [1/2,1] works fine too:

H(s,0) = G(2s-1,0) = g(2s-1)=h(s)

H(s,1) = G(2s-1,1)= g'(2s-1) = h'(s)

H(1,t) = G(1,t)= x_2

Thus, H is indeed the required path homotopy between f*g and f’*g’. * is almost like a binary operation for a group. The only difference is that [f]*[g] is not defined for every pair of classes, but only for those pairs [f], [g] for which f(1) = g(0), i.e. the end point of f is the starting point of g.

## Homotopy of Paths

For this post we will explain what is a homotopy of paths.

The book above is a nice introductory book on Topology, which includes a section of introductory Algebraic Topology.

Definition: If f and f’ are continuous maps of the space X into the space Y, we say that f is homotopic to f’ if there is a continuous F: X x I -> Y such that

F(x, 0)=f(x) and F(x,1) = f'(x)

for each x. The map F is called a homotopy between f and f’. If f is homotopic to f’, we write $f \simeq f'$.

If f and f’ are two paths in X, there is a stronger relation, called path homotopy, which requires that the end points of the path remain fixed during the deformation. We write $f \simeq_p f'$ if f and f’ are path homotopic.

Next, we will prove that the relations $\simeq$ and $\simeq_p$ are equivalence relations.

If f is a path, we shall denote its path-homotopy equivalence class by [f].

Proof: We shall verify the properties of an equivalence relation, namely reflexivity, symmetry and transitivity.

Reflexivity:

Given f, it is rather easy to see that $f \simeq f$. The map F(x,t) is the required homotopy.

F(x,0)=f(x) and F(x,1)=f(x) is clearly satisfied.

If f is a path, then F is certainly a path homotopy, since f and f itself has the same initial point and final point.

Symmetry:

Next we shall show that given $f \simeq f'$, we have $f' \simeq f$. Let F be a homotopy between f and f’. We can then verify that G(x,t) = F(x, 1-t) is a homotopy between f’ and f.

G(x,0) = F(x, 1)=f’ (x)

G(x,1) = F(x, 0) = f(x)

Furthermore, if F is a path homotopy, so is G.

G(0,t)=F(0, 1-t) = $x_0$

G(1,t)=F(1,1-t) = $x_1$

Transitivity:

Next, suppose that $f \simeq f'$ and $f' \simeq f''$, we show that $f \simeq f''$. Let F be a homotopy between f and f’, and let F’ be a homotopy between f’ and f”. This time, we need to define a slightly more complicated homotopy G: X x I -> Y by the equation

$G(x,t) = \begin{cases} F(x,2t) &\text{for }t\in [0,\frac{1}{2}],\\ F'(x, 2t-1) &\text{for } t\in [\frac{1}{2}, 1].\end{cases}$

First, we need to check if the map G is well defined at t=1/2. When t=1/2, we have F(x,2t) = F(x,1)=f'(x) = F'(x,2t-1).

Because G is continuous on the two closed subsets X x [0, 1/2] and X x [1/2, 1] of XxI, it is continuous on all of X x I, by the pasting lemma.

Thus, we may see that G is the required homotopy between f and f”.

G(x,0)=F(x,0) = f(x)

G(x,1) = F’ (x, 1) = f”(x)

We can also check that if F and F’ are path homotopies, so is G.

G(0,t) = F(0, 2t) = $x_0$

G(1, t)=F'(1, 2t-1) = $x_1$


text:
CaltechTHESIS
A Caltech Library Service

# The physics of ultracold neutrons and Fierz interference in beta decay

## Citation

Hickerson, Kevin Peter (2013) The physics of ultracold neutrons and Fierz interference in beta decay. Dissertation (Ph.D.), California Institute of Technology. http://resolver.caltech.edu/CaltechTHESIS:11022012-135115177

## Abstract

In the first component of this thesis, we investigate the physics of ultracold neutrons (UCN). UCN are neutrons so cold they can be stored inside of material, magnetic and gravitational bottles. Using this property we use UCN nonimaging optics to design a type of reflector that directs UCN upward in to vertical paths. Next we examine UCN passing through thin, multilayered foils. In the remaining sections we investigate the so-called Fierz interference term of free neutron beta decay, denoted $b_n$. It is theorized that contributions to scalar and tensor interactions from physics beyond the Standard Model could be detectable in the spectrum of neutron beta decay, manifest as a nonzero value for $b_n$. We investigate three techniques for measuring $b_n$. The first is to use the primordial helium abundance fraction and compare that to predictive Big Bang nucleosynthesis calculations. Second we extract $b_n$ from the spectral shape generated by the 2010 data set of the UCNA experiment. Third, we discuss progress toward constructing the UCNb experimental prototype. We present the design of this new experiment that uses the UCN source at LANSCE for measuring $b_n$, in which UCN are guided into a shielded $4\pi$ calorimeter where they are stored and decay. From Big Bang nucleosynthesis we can place the limit $0.021 < b_n < 0.277$ (90\% {C.L.}) on the neutron Fierz term and from the UCNA 2010 data we set $-0.044 < b_n < 0.218$ (90\% {C.L.}).

Item Type: Thesis (Dissertation (Ph.D.)) Ultracold neutrons, Fierz Interference California Institute of Technology Physics, Mathematics and Astronomy Physics Public (worldwide access) Filippone, Bradley W. Filippone, Bradley W. (chair)Hirata, Christopher M.Golwala, SunilLibbrecht, Kenneth GeorgeWise, Mark B. 9 October 2012 CaltechTHESIS:11022012-135115177 http://resolver.caltech.edu/CaltechTHESIS:11022012-135115177 No commercial reproduction, distribution, display or performance rights in this work are provided. 7250 CaltechTHESIS Kevin Hickerson 10 Dec 2012 21:42 09 Feb 2013 00:10

## Thesis Files

 Preview
PDF - Final Version
See Usage Policy.

10Mb

Repository Staff Only: item control page


text:
[NTG-context] Insufficient symbol fonts error

Alan Bowen acbowen at princeton.edu
Sun Oct 30 15:58:11 CET 2005

Giuseppe—

I am not using either the amsl or the nath modules. Never had to
before. (The file has compiled successfully in its current form, but
that was about a week ago.)

If I should be using one of these modules, which do you recommend?

Alan

On Oct 30, 2005, at 9:43 AM, Giuseppe Bilotta wrote:

> Sunday, October 30, 2005 Alan Bowen wrote:
>
>
>> I am getting some puzzling errors when I try to use the Greek fonts
>> in math formulae (to represent Greek numerals as written in mss.)
>>
>
>
>> In a short file, both
>> \overbar{\grk{kj}} and
>> $\overline{\hbox{\grk{kj}}}$
>> work as expected, even when they occur in Greek text.
>>
>
>
>> But in a long, complicated file with lots of Greek etc. and  the same
>> environment file, I get
>>
>
>
>> ! Math formula deleted: Insufficient symbol fonts.
>>
>
> Are you using the amsl or nath modules?
>
> --
> Giuseppe "Oblomov" Bilotta
>
> _______________________________________________
> ntg-context mailing list
> ntg-context at ntg.nl
> http://www.ntg.nl/mailman/listinfo/ntg-context
>


text:
# 2nd Order Linear Homogeneous Differential Equations 3

Let's use some initial conditions to solve for the particular solution


text:
# National Physical Laboratory of India

Agency overview File:Logo NPL india.svg 4 January 1947 New Delhi Prof.Dr. Venu Gopal Achanta, Director Council of Scientific and Industrial Research nplindia.org

The CSIR- National Physical Laboratory of India, situated in New Delhi, is the measurement standards laboratory of India. It maintains standards of SI units in India and calibrates the national standards of weights and measures.

## History of measurement systems in India

In the Harappan era, which is nearly 5000 years old, one finds excellent examples of town planning and architecture. The sizes of the bricks were the same all over the region. In the time of Chandragupta Maurya, some 2400 years ago, there was a well - defined system of weights and measures. The government of that time ensured that everybody used the same system. In the Indian medical system, Ayurveda, the units of mass and volume were well defined.

The measurement system during the time of the Mughal emperor, Akbar, the guz was the measure of length. The guz was widely used till the introduction of the metric system in India in 1956. During the British period, efforts were made to achieve uniformity in weights and measures. A compromise was reached in the system of measurements which continued till India's independence in 1947. After independence in 1947, it was realized that for fast industrial growth of the country, it would be necessary to establish a modern measurement system in the country. The Lok Sabha in April 1955 resolved : This house is of the opinion that the Government of India should take necessary steps to introduce uniform weights and measures throughout the country based on metric system.[1][circular reference]

## History of the National Physical Laboratory, India

The National Physical Laboratory, India was one of the earliest national laboratories set up under the Council of Scientific & Industrial Research. Jawaharlal Nehru laid the foundation stone of NPL on 4 January 1947. Dr. K. S. Krishnan was the first Director of the laboratory. The main building of the laboratory was formally opened by Former Deputy Prime Minister Sardar Vallabhbhai Patel on 21 January 1950. Former Prime Minister Indira Gandhi, inaugurated the Silver Jubilee Celebration of the Laboratory on 23 December 1975.

NPL Charter:-

The main aim of the laboratory is to strengthen and advance physics-based research and development for the overall development of science and technology in the country. In particular its objectives are:

To establish, maintain and improve continuously by research, for the benefit of the nation, National Standards of Measurements and to realize the Units based on International System (Under the subordinate Legislations of Weights and Measures Act 1956, reissued in 1988 under the 1976 Act). To identify and conduct after due consideration, research in areas of physics which are most appropriate to the needsof the nation and for advancement of field

To assist industries, national and other agencies in their developmental tasks by precision measurements, calibration, development of devices, processes, and other allied problems related to physics.

To keep itself informed of and study critically the status of physics.

File:Traceability Pyramid at NPL.png
Newly established structures at NPL campus

## Maintenance of standards of measurements in India

Each modernized country, including India has a National Metrological Institute (NMI), which maintains the standards of measurements. This responsibility has been given to the National Physical Laboratory, New Delhi.

### Metre

The standard unit of length, metre, is realized by employing a stabilized helium-neon laser as a source of light. Its frequency is measured experimentally. From this value of frequency and the internationally accepted value of the speed of light (Lua error in package.lua at line 80: module 'Module:Val/units' not found.), the wavelength is determined using the relation:

$wavelength = \frac{velocity-of-light}{frequency}$

The nominal value of wavelength, employed at NPL is 633 nanometer. By a sophisticated instrument, known as an optical interferometer, any length can be measured in terms of the wavelength of laser light.

The present level of uncertainty attained at NPL in length measurements is ±3 × 10−9. However, in most measurements, an uncertainty of ±1 × 10−6 is adequate.

### Kilogramme

The Indian national standard of mass, kilogramme, is copy number 57 of the international prototype of the kilogram supplied by the International Bureau of Weights and Measures (BIPM: French – Bureau International des Poids et Mesures), Paris. This is a platinum-iridium cylinder whose mass is measured against the international prototype at BIPM. The NPL also maintains a group of transfer standard kilograms made of non-magnetic stainless steel and nickel-chromium alloy.

The uncertainty in mass measurements at NPL is ±4.6 × 10−9.

### Second

The national standard of time interval, second as well as frequency, is maintained through four parameters, which can be measured most accurately. Therefore, attempts are made to link other physical quantities to time and frequency. The standard maintained at NPL has to be linked to different users. This process, known as dissemination, is carried out in a number of ways. For applications requiring low levels of uncertainty, there is satellite based dissemination service, which utilizes the Indian national satellite, INSAT. Time is also disseminated through TV, radio, and special telephone services. The caesium atomic clocks maintained at NPL are linked to other such instituted all over the world through a set of global positioning satellites.

### Ampere

The unit of electric current, ampere, is realized at NPL by measuring the volt and the ohm separately.

The uncertainty in measurement of ampere is ± 1 × 10−6.

### Kelvin

The standard of temperature is based on the International Temperature Scale of 1990 (ITS-90). This is based on the assigned temperatures to several fixed points. One of the most fundamental temperatures of these is the triple point of water. At this temperature, ice, water and steam are at equilibrium with each other. This temperature has been assigned the value of 273.16 kelvins. This temperature can be realized, maintained and measured in the laboratory. At present temperature standards maintained at NPL cover a range of 54 to 2,473 kelvins.

The uncertainty in its measure is ± 2.5 × 10−4.

### Candela

The unit of luminous intensity, candela, is realized by using an absolute radiometer. For practical work, a group of tungsten incandescent lamps are used.

The level of uncertainty is ± 1.3 × 10−2.

### Mole

Experimental work has been initiated to realize mole, the SI unit for amount of substance

The NPL does not maintain standards of measurements for ionizing radiations. This is the responsibility of the Homi Bhabha Atomic Research Centre, Mumbai.

## Calibrator of weights and measures

The standards maintained at NPL are periodically compared with standards maintained at other National Metrological Institutes in the world as well as the BIPM in Paris. This exercise ensures that Indian national standards are equivalent to those of the rest of the world.

Any measurement made in a country should be directly or indirectly linked to the national standards of the country, For this purpose, a chain of laboratories are set up in different states of the country. The weights and measures used in daily life are tested in the laboratories and certified. It is the responsibility of the NPL to calibrate the measurement standards in these laboratories at different levels. In this manner, the measurements made in any part of the country are linked to the national standards and through them to the international standards.

The weights and balances used in local markets and other areas are expected to be certified by the Department of Weights and Measures of the local government. Working standards of these local departments should, in turn, be calibrated against the state level standards or any other laboratory which is entitled to do so. The state level laboratories are required to get their standards calibrated from the NPL at the national level which is equivalent to the international standards.

## Bharatiya Nirdeshak Dravya (BND) or Indian Reference Materials

Bharatiya Nirdeshak Dravya (BND) or Indian reference materials are reference materials developed by NPL which derive their traceability from National Standards.

## Research

NPL is also involved in research. One of the important research activities undertaken by NPL is to devise the chemical formula for the indelible ink which is being used in the Indian elections to prevent fraudulent voting. This ink, manufactured by the Mysore Paints and Varnish Limited is applied on the finger nail of the voter as an indicator that the voter has already cast his vote.

NPL also have section working on development of biosensors. Currently the Biomedical Instrumentation section is headed by Dr. R. K. Kotnala and section is primarily focusing on development of sensor for cholesterol, measurement and microfluidic based biosensors. Section is also developing biosensors for Uric acid detection.

## NPL's Contributions

### The Indelible Mark/Ink

During general election, nearly 40 million people wear a CSIR mark on their fingers. The Indelible ink used to mark the fingernail of a Voter during general elections is a time-tested gift of CSIR to the spirit of democracy. Developed in 1952, it was first produced in-campus. Subsequently, industry has been manufacturing the Ink. It is also exported to Sri Lanka, Indonesia, Turkey and other democracies.

### Pristine Air-Quality Monitoring Station at Palampur

National Physical Laboratory (NPL) has established an atmospheric monitoring station in the campus of Institute of Himalayan Bioresource Technology (IHBT) at Palampur (H.P.) at an altitude of 1391 m for generating the base data for atmospheric trace species & properties to serve as reference for comparison of polluted atmosphere in India. At this station, NPL has installed state of art air monitoring system, greenhouse gas measurement system and Raman Lidar. A number of parameters like CO, NO, NO2, NH3, SO2, O3, PM, HC & BC besides CO2 & CH4 are being currently monitored at this station which is also equipped with weather station (AWS) for measurement of weather parameters.[2][3]

### Gold Standard (BND-4201)

The BND-4201 is first Indian reference material for gold of ‘9999’ fineness (gold that is 99.99% pure with impurities of only 100 parts-per-million).

## References

1. Indian units of measurement
2. "National Physical Laboratory(NPL)- CSIR dedicates the first "Pristine air-quality monitoring station at Palampur" to the Nation". pib.nic.in.
3. "CSIR-NPL launches India's First Pristine Air-Quality Monitoring Station at Palampur". MyGov Blogs. 25 March 2017.


text:
# ?SPAM? - [texhax] superscripts and spacing problem

Philip TAYLOR P.Taylor at Rhul.Ac.Uk
Mon Nov 28 18:11:15 CET 2005

Joey Coyle wrote:

> Hi,
>
> I am printing out musical chords like C7 or A7 with the 7
> superscripted...  It looks fine with the C7, but with the A7, with  the
> inward slope of the A, the 7 should be moved closer to the A...,  I know
> I can do this manually with \hspace, but I have a macro   \DomSev{C}  or
> \DomSev{A} that prints the chord and the macro doesn't  know what note
> it will take as an argument...
>
> I thought the beauty of Tex was that it handled this kind of thing
> automatically?

Only within the limits of the TeX paradigm, which is to treat
everything as if it were a rectangular box.  Once you want to
start looking /inside/ the box (in your case, to differentiate
between the kerning needed for A^7 v. C^7), things are a lot
harder, which is why kerning information such as this comes
from the font designer via the TFM file rather than being
left to TeX to work out for itself.  In your case, one
possible solution would be to compare the two widths

(1)	\wd \hbox {#1V}

and

(2)	\wd \hbox {#1{}V}

(These aren't real TeX constructs : you'd need a couple
of intermediate steps to get at these widths -- I can
deal with that later if you don't know how to proceed)

Then you could use the difference (1) - (2) to compute
the correction needed between $\hbox {A}^\hbox {7}$
w.r.t. that needed for $\hbox {C}^\hbox {7}$.

Sample code (compare the results visually)

\def \domsev #1%
{\setbox 0 = \hbox {#1{}V}%
\setbox 2 = \hbox {#1V}
\dimen 0 = \wd 0
\advance \dimen 0 by -\wd 2
\dimen 0 = 0 pt %%% pretend we did none of the above
$\hbox {#1}\kern -\dimen 0 ^7$
}

\def \DomSev #1%
{\setbox 0 = \hbox {#1{}V}%
\setbox 2 = \hbox {#1V}
\dimen 0 = \wd 0
\advance \dimen 0 by -\wd 2
$\hbox {#1}\kern -\dimen 0 ^7$
}

\domsev {A}
\domsev {C}

\DomSev {A}
\DomSev {C}

\end

Philip Taylor


text:
# Using 'xfrac' with 'mathrsfs'

I'm attempting to move from the nicefrac package to xfrac. The following source compiles, but with a warning:

\RequirePackage{fix-cm}
\documentclass{article}
\usepackage{xfrac}
\usepackage{mathrsfs}
\begin{document}
$\sfrac{1}{2}$
\end{document}

LaTeX Font Warning: Font shape 'U/rsfs/m/n' in size <3.49998> not available

It appears as though xfrac attempts to load all current math alphabets in its typeset size, even if they are not used in the fraction. Is there a variant of rsfs that supports fractional sizes? Or a way to prevent xfrac from loading it unless it is needed?

Basically, I'd like to suppress this warning unless LaTeX is actually typesetting a glyph from rsfs at a fractional size. Is this possible?

Fixed font sizes of rsfs font in NFSS scheme is declared in ursfs.fd. It contains

\ProvidesFile{ursfs.fd}[1998/03/24 rsfs font definition file (jk)]
\DeclareFontFamily{U}{rsfs}{\skewchar\font127 }
\DeclareFontShape{U}{rsfs}{m}{n}{%
<5> <6> rsfs5
<7> rsfs7
<8> <9> <10> <10.95> <12> <14.4> <17.28> <20.74> <24.88> rsfs10
}{}

Now you can modify it for continuous font sizes. You can just put the code in preamble:

\RequirePackage{fix-cm}
\documentclass{article}
\usepackage{xfrac}
\usepackage{mathrsfs}
\DeclareFontFamily{U}{rsfs}{\skewchar\font127 }
\DeclareFontShape{U}{rsfs}{m}{n}{%
<-6> rsfs5
<6-8> rsfs7
<8-> rsfs10
}{}
\begin{document}
$\sfrac{1}{2}$
\end{document}

• Hah, you barely beat me to it! I just discovered this by digging into fix-cm's source. I was, however, missing the \skewchar\font127. Is there any reason not to define this in terms of continuous font sizes? Or is this just something that wasn't possible long ago and has simply propagated through today? – Matt B. Feb 9 '11 at 19:03
• Now we are using Type1 fonts. They are scalable vector font. However, there are only bitmap fonts produced by MetaFont years ago, scaling is not preferred for performance reason. – Leo Liu Feb 9 '11 at 19:45


text:
# URL issues in bibliography

I am having problems building my bibliography and would like some advice. The main issues are:

1. 'URL address' and 'date accessed' not always included in the bibliography entries for websites.
2. URL addresses in the bibliography not always in line with other text (please see screenshot).
3. Bibliography entries are not always numbered sequentially in the article and bibliography.
4. Maintain same style for referening articles (as shown in the example).

I have tried several bibliography styles and looked through similar posts on tex.stackexchange but none seem to solve all of the problems in one go i.e. in the screenshot example, only 3. is solved but in other attempts, it will be 1. etc...I also tried using Biblatex and Biber but I get the following error message:

did you pass the "backend=biber" option to BibLaTeX?

I am using pdfLaTeX+MakeIndex+BibteX typeset to compile my document. The .bib library file is created using Mendeley. I attach my Bibtex, input code and a screenshot of the output.

Bibtex

@misc{Car2018,
title = {{Carillion fallout: 'They've literally locked the gate' - BBC News}},
urldate = {2018-01-15}

@misc{Wik2018,
title = {{List of software for Monte Carlo molecular modeling - Wikipedia}},
url = {https://en.wikipedia.org/wiki/List{\_}of{\_}software{\_}for{\_}Monte{\_}Carlo{\_}molecular{\_}modeling},
urldate = {2018-01-15}
}

@article{McKenna2010,
abstract = {Next-generation DNA sequencing (NGS) projects, such as the 1000 Genomes Project, are already revolutionizing our understanding of genetic variation among individuals. However, the massive data sets generated by NGS--the 1000 Genome pilot alone includes nearly five terabases--make writing feature-rich, efficient, and robust analysis tools difficult for even computationally sophisticated individuals. Indeed, many professionals are limited in the scope and the ease with which they can answer scientific questions by the complexity of accessing and manipulating the data produced by these machines. Here, we discuss our Genome Analysis Toolkit (GATK), a structured programming framework designed to ease the development of efficient and robust analysis tools for next-generation DNA sequencers using the functional programming philosophy of MapReduce. The GATK provides a small but rich set of data access patterns that encompass the majority of analysis tool needs. Separating specific analysis calculations from common data management infrastructure enables us to optimize the GATK framework for correctness, stability, and CPU and memory efficiency and to enable distributed and shared memory parallelization. We highlight the capabilities of the GATK by describing the implementation and application of robust, scale-tolerant tools like coverage calculators and single nucleotide polymorphism (SNP) calling. We conclude that the GATK programming framework enables developers and analysts to quickly and easily write efficient and robust NGS tools, many of which have already been incorporated into large-scale sequencing projects like the 1000 Genomes Project and The Cancer Genome Atlas.},
author = {McKenna, Aaron and Hanna, Matthew and Banks, Eric and Sivachenko, Andrey and Cibulskis, Kristian and Kernytsky, Andrew and Garimella, Kiran and Altshuler, David and Gabriel, Stacey and Daly, Mark and DePristo, Mark A},
doi = {10.1101/gr.107524.110},
issn = {1549-5469},
journal = {Genome research},
month = {sep},
number = {9},
pages = {1297--303},
pmid = {20644199},
publisher = {Cold Spring Harbor Laboratory Press},
title = {{The Genome Analysis Toolkit: a MapReduce framework for analyzing next-generation DNA sequencing data.}},
url = {http://www.ncbi.nlm.nih.gov/pubmed/20644199 http://www.pubmedcentral.nih.gov/articlerender.fcgi?artid=PMC2928508},
volume = {20},
year = {2010}
}

Input code (using MikTex 2.9, Texworks)

\documentclass[a4paper,11pt]{article}
\usepackage{color}
\usepackage{graphicx}
\usepackage{times}
\usepackage[numbers]{natbib}
\bibliographystyle{unsrtnat}

\begin{document}

Webpage test 1 \cite{Car2018}.
Webpage test 2 \cite{Wik2018}.
Article test 1 \cite{McKenna2010}

\bibliography{library}

\end{document}

I know there are several problems here but I'm not sure which one is attributed to Mendeley or TexWorks which is why I have put them in one post. Any advice you can give would be great!

Use url. This is a version of your source

\documentclass[a4paper,11pt]{article}
\usepackage{color}
\usepackage{graphicx}
\usepackage{times}
\usepackage[numbers]{natbib}
\bibliographystyle{unsrtnat}

\usepackage[hyphens]{url}
\usepackage[T1]{fontenc}

\begin{document}

Webpage test 1 \cite{Car2018}.
Webpage test 2 \cite{Wik2018}.
Article test 1 \cite{McKenna2010}

\bibliography{library}

\end{document}

The unsrtnat style does not make use of the urldate entry. But you could use note instead

 @misc{Car2018,
title = {{Carillion fallout: 'They've literally locked the gate' - BBC News}},
note = {Visited: 2018-01-15}
}

@misc{Wik2018,
title = {{List of software for Monte Carlo molecular modeling - Wikipedia}},
url = {https://en.wikipedia.org/wiki/List_of_software_for_Monte_Carlo_molecular_modeling},
note = {Visited: 2018-01-15}
}

@article{McKenna2010,
abstract = {Next-generation DNA sequencing (NGS) projects, such as the 1000 Genomes Project, are already revolutionizing our understanding of genetic variation among individuals. However, the massive data sets generated by NGS--the 1000 Genome pilot alone includes nearly five terabases--make writing feature-rich, efficient, and robust analysis tools difficult for even computationally sophisticated individuals. Indeed, many professionals are limited in the scope and the ease with which they can answer scientific questions by the complexity of accessing and manipulating the data produced by these machines. Here, we discuss our Genome Analysis Toolkit (GATK), a structured programming framework designed to ease the development of efficient and robust analysis tools for next-generation DNA sequencers using the functional programming philosophy of MapReduce. The GATK provides a small but rich set of data access patterns that encompass the majority of analysis tool needs. Separating specific analysis calculations from common data management infrastructure enables us to optimize the GATK framework for correctness, stability, and CPU and memory efficiency and to enable distributed and shared memory parallelization. We highlight the capabilities of the GATK by describing the implementation and application of robust, scale-tolerant tools like coverage calculators and single nucleotide polymorphism (SNP) calling. We conclude that the GATK programming framework enables developers and analysts to quickly and easily write efficient and robust NGS tools, many of which have already been incorporated into large-scale sequencing projects like the 1000 Genomes Project and The Cancer Genome Atlas.},
author = {McKenna, Aaron and Hanna, Matthew and Banks, Eric and Sivachenko, Andrey and Cibulskis, Kristian and Kernytsky, Andrew and Garimella, Kiran and Altshuler, David and Gabriel, Stacey and Daly, Mark and DePristo, Mark A},
doi = {10.1101/gr.107524.110},
issn = {1549-5469},
journal = {Genome research},
month = {sep},
number = {9},
pages = {1297--303},
pmid = {20644199},
publisher = {Cold Spring Harbor Laboratory Press},
title = {{The Genome Analysis Toolkit: a MapReduce framework for
analyzing next-generation DNA sequencing data.}},
url = {http://www.ncbi.nlm.nih.gov/pubmed/20644199
http://www.pubmedcentral.nih.gov/articlerender.fcgi?artid=PMC2928508},
volume = {20},
year = {2010}
}

• The note field does not appear in the bibtex file. When you say use note, do you mean manually inserting this field into the bibtex file created by Mendeley? Jan 15, 2018 at 20:27
• @Kevin You can either to that or look for a bibliography style that does make use of the urldate field Jan 15, 2018 at 20:30

After some searching Mendeley problem displaying URL visiting date with Biblatexand trial and error, I managed to resolve all the problems. Here's my source:

\documentclass[a4paper,11pt]{article}
\usepackage{color}
\usepackage{graphicx}
\usepackage{times}
\usepackage[british]{babel}
\usepackage[backend=bibtex, citestyle=ieee, bibstyle=ieee]{biblatex}

\begin{document}

Webpage test 1 \cite{Car2018}.
Webpage test 2 \cite{Wik2018}.
Article test 1 \cite{McKenna2010}

\printbibliography

\end{document}

And the output:

To resolve problem (1), I used bibliography style ieee and biblatex. To resolve problem (2), I deleted the 'URL' from Mendeley and kept the 'DOI'. Problems (3) and (4) appear to correct themselves when using the ieee bib. style.

• Note that the output shows that you should not escape special characters in the url field. URLs should be given as is without change. biblatex can deal with special characters without a problem. Jan 16, 2018 at 10:26


text:
# Send funds to third parties

Send funds to third parties, such as suppliers or utility providers.

Users that have a sendToThirdParty capability allowed and enabled can make business-related fund transfers to third parties, such as their suppliers or providers.

Third-party accounts are external accounts not owned by your user and therefore not connected to their legal entity as a transfer instrument. Currently, Adyen only supports sending funds to third-party bank accounts. In the future, you will be able to send to third-party wallets and cards.

## Before you begin

Before your user can send funds to third parties, you must:

• Reach out to your Adyen contact to:
• Enable transfers for the source balance account.
• Make sure that the account holder has the sendToThirdParty capability allowed and enabled.
• Implement Adyen's Authentication SDK for transfers in the European Economic Area (EEA).

## Send funds to third parties

When sending funds to third-party bank accounts, the required bank account details depend on bank identification type. There are also additional required fields if you're doing a cross-border transfer.

To start a transfer, make a POST /transfers request specifying:

Parameter name Required Description
balanceAccountId The ID of the balance account from which funds are deducted.
category Set to bank.
counterparty.bankAccount Contains details about the bank account and its owner.
• The accountHolder.fullName is always required. For cross-border transfers, the accountHolder.address object is also required.
• The accountIdentification is required. The fields that you need to send in this object are determined by the accountIdentification.type. See bank account identification types.
priority The priority of the bank transfer, which affects the speed of the transfer and the fees you have to pay. The possible values are regular, fast, or wire.
For cross-border transfers, set the priority to wire.
description Your description for the transfer. See here for allowed characters.
reference Your reference for the transfer. This is only used within your platform and not sent to the recipient. If you don't provide this in the request, Adyen generates a unique reference.
referenceForBeneficiary Your reference for the transfer that Adyen sends to the recipient.

Select a tab below for examples.

Here is an example of a local funds transfer of EUR funds to an IBAN bank account. When providing the details of the bank account, set the accountIdentification.type to iban.

For possible values for local transfers, see Bank account identification types.

Local funds transfer to an IBAN bank account
curl https://balanceplatform-api-test.adyen.com/btl/v3/transfers \
-H "content-type: application/json" \
-H "X-API-Key: YOUR_API_KEY" \
-X POST \
-d '{
"amount": {
"currency": "EUR",
"value": 150000
},
"category": "bank",
"priority": "regular",
"balanceAccountId": "BA00000000000000000000001",
"counterparty": {
"bankAccount": {
"accountHolder": {
"fullName": "A. Klaassen"
},
"accountIdentification": {
"type": "iban",
"iban": "NL91ABNA0417164300"
}
}
},
"referenceForBeneficiary": "YOUR_REFERENCE_SENT_TO_THE_BENEFICIARY",
"reference": "YOUR_INTERNAL_REFERENCE",
"description": "YOUR_DESCRIPTION"
}'

If the transfer request is successful, you receive an HTTP 200 OK response containing an id of the transfer request. Adyen informs your server of the status of the transfer through webhooks.

## Bank account identification types

When transferring funds to a third-party bank account, you need to provide the accountIdentification.type in your API request. The type also determines the required bank account details.

1. To determine the accountIdentification.type value, you must have:
• The country of the recipient bank account.
• The currency of the funds you are transferring.

2. Find the combination in the table below and select the type to see the required fields.

This combination also identifies if you are doing a local or cross-border transfer. Cross-border transfers must be sent as wire transfers, and you must include the address of the bank account owner.

Country of recipient bank account Currency of funds transfer Type Notes Character limits for description
Single Euro Payments Area (SEPA) countries EUR iban Maximum 140 characters. Allowed characters: / - ? : ( ) . , ' + Space
United States USD usLocal Allowed characters: / - ? : ( ) . , ' + Space
United Kingdom GBP ukLocal Maximum 140 characters. Allowed characters: / - ? : ( ) . , ' + Space
Poland PLN plLocal or iban Maximum 140 characters. Allowed characters: / - ? : ( ) . , ' + Space
Czech Republic CZK czLocal or iban Maximum 140 characters. Allowed characters: / - ? : ( ) . , ' + Space
Sweden SEK seLocal Maximum 140 characters. Allowed characters: / - ? : ( ) . , ' + Space
Countries where IBAN is required Any iban This is a cross-border transfer. Make sure to:
• Set the priority to wire.
• Include the address of the bank owner.
Maximum 140 characters. Allowed characters: [a-z][A-Z][0-9] . , – ( ) / = ' + : ? ! ” % & * < > ; Space
Countries where IBAN is optional
(for example, Poland or Czech Republic)
Any iban or numberAndBic This is a cross-border transfer. Make sure to:
• Set the priority to wire.
• Include the address of the bank owner.
Maximum 140 characters. Allowed characters: [a-z][A-Z][0-9] . , – ( ) / = ' + : ? ! ” % & * < > ; Space
Other countries not listed above Any numberAndBic This is a cross-border transfer. Make sure to:
• Set the priority to wire.
• Include the address of the bank owner.
Maximum 140 characters. Allowed characters: [a-z][A-Z][0-9] . , – ( ) / = ' + : ? ! ” % & * < > ; Space

The table below shows the required fields that you need to send in the accountIdentification object based on the type.
iban iban
usLocal accountNumber, routingNumber
ukLocal accountNumber, sortCode
plLocal accountNumber
czLocal accountNumber, bankCode
seLocal accountNumber, clearingNumber
numberAndBic accountNumber, bic


text:
# How did they do this?

1. Oct 20, 2013

### Superposed_Cat

How did they did they do the second line? Any help appreciated, thanks in advance.

#### Attached Files:

• ###### eq0020MP.gif
File size:
3.9 KB
Views:
71
2. Oct 20, 2013

### Emphacy

You know the rules of integration right? Imagine differentiating $x^3$. You get $3x^2$.
Now reverse the differentiation.

So integrating $3x^2$ you first increase the power and then divide by the new power. So...

1) Raise the power: $3x^3$
2) Divide by the new power: $\frac{3}{3}x^3$ = $x^3$

That's all that's happening on the second line.

3. Oct 20, 2013

### gopher_p

$\int 3x^\frac{3}{4}+7x^{-5}+\frac{1}{6}x^{-\frac{1}{2}}\ dx=\int 3x^\frac{3}{4}\ dx+\int 7x^{-5}\ dx+\int \frac{1}{6}x^{-\frac{1}{2}}\ dx$ using the sum rule for integrals

$\int 3x^\frac{3}{4}\ dx+\int 7x^{-5}\ dx+\int \frac{1}{6}x^{-\frac{1}{2}}\ dx=3\int x^\frac{3}{4}\ dx+7\int x^{-5}\ dx+\frac{1}{6}\int x^{-\frac{1}{2}}\ dx$ using the constant multiple rule for integrals

$3\int x^\frac{3}{4}\ dx+7\int x^{-5}\ dx+\frac{1}{6}\int x^{-\frac{1}{2}}\ dx=3\cdot\frac{1}{\frac{3}{4}+1} x^{\frac{3}{4}+1}+7\cdot\frac{1}{-5+1}x^{-5+1}+\frac{1}{6}\cdot\frac{1}{-\frac{1}{2}+1} x^{-\frac{1}{2}+1}+c$ using the integral formula $\int x^\alpha\ dx=\frac{1}{a+1}x^{\alpha+1}+c$ when $\alpha\neq-1$

4. Oct 20, 2013

### Superposed_Cat

Profuse thanks dear sir.


text:
- Current Issue - Ahead of Print Articles - All Issues - Search - Open Access - Information for Authors - Downloads - Guideline - Regulations ㆍPaper Submission ㆍPaper Reviewing ㆍPublication and Distribution - Code of Ethics - For Authors ㆍOnlilne Submission ㆍMy Manuscript - For Reviewers - For Editors
 Exponential rank of extensions of $C^*$-algebras Bull. Korean Math. Soc. 1997 Vol. 34, No. 3, 395-401 Ja A Jeong and Gie Hyun Park Kyung Hee University, Hanshin University Abstract : We show that if $I$ is an ideal of a $C^*$-algebra $A$ such that the unitary group of $\tilde I$ is connected then $cer(A)\leq cer(I)+cer(A/I)$, where $cer(A)$ denotes the $C^{*}$-exponential rank of $A$. Keywords : exponential rank, real rank MSC numbers : 46L05 Downloads: Full-text PDF