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# Fourier-transform ion cyclotron resonance

Acronym FTICR Mass spectrometry Ion trapQuadrupole ion trapPenning trapOrbitrap

Fourier-transform ion cyclotron resonance mass spectrometry is a type of mass analyzer (or mass spectrometer) for determining the mass-to-charge ratio (m/z) of ions based on the cyclotron frequency of the ions in a fixed magnetic field.[1] The ions are trapped in a Penning trap (a magnetic field with electric trapping plates), where they are excited (at their resonant cyclotron frequencies) to a larger cyclotron radius by an oscillating electric field orthogonal to the magnetic field. After the excitation field is removed, the ions are rotating at their cyclotron frequency in phase (as a "packet" of ions). These ions induce a charge (detected as an image current) on a pair of electrodes as the packets of ions pass close to them. The resulting signal is called a free induction decay (FID), transient or interferogram that consists of a superposition of sine waves. The useful signal is extracted from this data by performing a Fourier transform to give a mass spectrum.

## History

FT-ICR was invented by Melvin B. Comisarow[2] and Alan G. Marshall at the University of British Columbia. The first paper appeared in Chemical Physics Letters in 1974.[3] The inspiration was earlier developments in conventional ICR and Fourier-transform nuclear magnetic resonance (FT-NMR) spectroscopy. Marshall has continued to develop the technique at The Ohio State University and Florida State University.

## Theory

Linear ion trap – Fourier-transform ion cyclotron resonance mass spectrometer (panels around magnet are missing)

The physics of FTICR is similar to that of a cyclotron at least in the first approximation.

In the simplest idealized form, the relationship between the cyclotron frequency and the mass-to-charge ratio is given by

${\displaystyle f={\frac {qB}{2\pi m}},}$

where f = cyclotron frequency, q = ion charge, B = magnetic field strength and m = ion mass.

This is more often represented in angular frequency:

${\displaystyle \omega _{\text{c}}={\frac {qB}{m}},}$

where ${\displaystyle \omega _{\text{c}}}$ is the angular cyclotron frequency, which is related to frequency by the definition ${\displaystyle f={\frac {\omega }{2\pi }}}$.

Because of the quadrupolar electrical field used to trap the ions in the axial direction, this relationship is only approximate. The axial electrical trapping results in axial oscillations within the trap with the (angular) frequency

${\displaystyle \omega _{\text{t}}={\sqrt {\frac {q\alpha }{m}}},}$

where ${\displaystyle \alpha }$ is a constant similar to the spring constant of a harmonic oscillator and is dependent on applied voltage, trap dimensions and trap geometry.

The electric field and the resulting axial harmonic motion reduces the cyclotron frequency and introduces a second radial motion called magnetron motion that occurs at the magnetron frequency. The cyclotron motion is still the frequency being used, but the relationship above is not exact due to this phenomenon. The natural angular frequencies of motion are

${\displaystyle \omega _{\pm }={\frac {\omega _{\text{c}}}{2}}\pm {\sqrt {\left({\frac {\omega _{\text{c}}}{2}}\right)^{2}-{\frac {\omega _{\text{t}}^{2}}{2}}}},}$

where ${\displaystyle \omega _{\text{t}}}$ is the axial trapping frequency due the axial electrical trapping and ${\displaystyle \omega _{+}}$ is the reduced cyclotron (angular) frequency and ${\displaystyle \omega _{-}}$ is the magnetron (angular) frequency. Again, ${\displaystyle \omega _{+}}$ is what is typically measured in FTICR. The meaning of this equation can be understood qualitatively by considering the case where ${\displaystyle \omega _{\text{t}}}$ is small, which is generally true. In that case the value of the radical is just slightly less than ${\displaystyle \omega _{\text{c}}/2}$, and the value of ${\displaystyle \omega _{+}}$ is just slightly less than ${\displaystyle \omega _{\text{c}}}$ (the cyclotron frequency has been slightly reduced). For ${\displaystyle \omega _{-}}$ the value of the radical is the same (slightly less than ${\displaystyle \omega _{\text{c}}/2}$), but it is being subtracted from ${\displaystyle \omega _{\text{c}}/2}$, resulting in a small number equal to ${\displaystyle \omega _{\text{c}}-\omega _{+}}$ (i.e. the amount that the cyclotron frequency was reduced by).

## Instrumentation

FTICR-MS differs significantly from other mass spectrometry techniques in that the ions are not detected by hitting a detector such as an electron multiplier but only by passing near detection plates. Additionally the masses are not resolved in space or time as with other techniques but only by the ion cyclotron resonance (rotational) frequency that each ion produces as it rotates in a magnetic field. Thus, the different ions are not detected in different places as with sector instruments or at different times as with time-of-flight instruments, but all ions are detected simultaneously during the detection interval. This provides an increase in the observed signal-to-noise ratio owing to the principles of Fellgett's advantage.[1] In FTICR-MS, resolution can be improved either by increasing the strength of the magnet (in teslas) or by increasing the detection duration.[4]

### Cells

A cylindrical ICR cell. The walls of the cell are made of copper, and ions enter the cell from the right, transmitted by the octopole ion guides.

A review of different cell geometries with their specific electric configurations is available in the literature.[5] However, ICR cells can belong to one of the following two categories: closed cells or open cells.

Several closed ICR cells with different geometries were fabricated and their performance has been characterized. Grids were used as end caps to apply an axial electric field for trapping ions axially (parallel to the magnetic field lines). Ions can be either generated inside the cell or can be injected to the cell from an external ionization source. Nested ICR cells with double pair of grids were also fabricated to trap both positive and negative ions simultaneously.

The most common open cell geometry is a cylinder, which is axially segmented to produce electrodes in the shape of a ring. The central ring electrode is commonly used for applying radial excitation electric field and detection. DC electric voltage is applied on the terminal ring electrodes to trap ions along the magnetic field lines.[6] Open cylindrical cells with ring electrodes of different diameters have also been designed.[7] They proved not only capable in trapping and detecting both ion polarities simultaneously, but also they succeeded to separate positive from negative ions radially. This presented a large discrimination in kinetic ion acceleration between positive and negative ions trapped simultaneously inside the new cell. Several ion axial acceleration schemes were recently written for ion–ion collision studies.[8]

### Stored-waveform inverse Fourier transform

Stored-waveform inverse Fourier transform (SWIFT) is a method for the creation of excitation waveforms for FTMS.[9] The time-domain excitation waveform is formed from the inverse Fourier transform of the appropriate frequency-domain excitation spectrum, which is chosen to excite the resonance frequencies of selected ions. The SWIFT procedure can be used to select ions for tandem mass spectrometry experiments.

## Applications

Fourier-transform ion cyclotron resonance (FTICR) mass spectrometry is a high-resolution technique that can be used to determine masses with high accuracy. Many applications of FTICR-MS use this mass accuracy to help determine the composition of molecules based on accurate mass. This is possible due to the mass defect of the elements. FTICR-MS is able to achieve higher levels of mass accuracy than other forms of mass spectrometer, in part, because a superconducting magnet is much more stable than radio-frequency (RF) voltage.[10] Another place that FTICR-MS is useful is in dealing with complex mixtures, such as biomass or waste liquefaction products,[11][12] since the resolution (narrow peak width) allows the signals of two ions with similar mass-to-charge ratios (m/z) to be detected as distinct ions.[13][14][15] This high resolution is also useful in studying large macromolecules such as proteins with multiple charges, which can be produced by electrospray ionization. For example, attomole level of detection of two peptides has been reported.[16] These large molecules contain a distribution of isotopes that produce a series of isotopic peaks. Because the isotopic peaks are close to each other on the m/z axis, due to the multiple charges, the high resolving power of the FTICR is extremely useful. FTICR-MS is very useful in other studies of proteomics as well. It achieves exceptional resolution in both top-down and bottom-up proteomics. Electron-capture dissociation (ECD), collisional-induced dissociation (CID), and infrared multiphoton dissociation (IRMPD) are all utilized to produce fragment spectra in tandem mass spectrometry experiments.[17] Although CID and IRMPD use vibrational excitation to further dissociate peptides by breaking the backbone amide linkages, which are typically low in energy and weak, CID and IRMPD may also cause dissociation of post-translational modifications. ECD, on the other hand, allows specific modifications to be preserved. This is quite useful in analyzing phosphorylation states, O- or N-linked glycosylation, and sulfating.[17]

## References

1. ^ a b Marshall, A. G.; Hendrickson, C. L.; Jackson, G. S. (1998). "Fourier transform ion cyclotron resonance mass spectrometry: a primer". Mass Spectrom. Rev. 17 (1): 1–35. Bibcode:1998MSRv...17....1M. doi:10.1002/(sici)1098-2787(1998)17:1<1::aid-mas1>3.0.co;2-k. PMID 9768511.
2. ^ "UBC Chemistry Personnel: Melvin B. Comisarow". University of British Columbia. Retrieved 2009-11-05.
3. ^ Comisarow, Melvin B. (1974). "Fourier transform ion cyclotron resonance spectroscopy". Chemical Physics Letters. 25 (2): 282–283. Bibcode:1974CPL....25..282C. doi:10.1016/0009-2614(74)89137-2.
4. ^ Marshall, A. (2002). "Fourier transform ion cyclotron resonance detection: principles and experimental configurations". International Journal of Mass Spectrometry. 215 (1–3): 59–75. Bibcode:2002IJMSp.215...59M. doi:10.1016/S1387-3806(01)00588-7.
5. ^ Guan, Shenheng; Marshall, Alan G. (1995). "Ion traps for Fourier transform ion cyclotron resonance mass spectrometry: principles and design of geometric and electric configurations". International Journal of Mass Spectrometry and Ion Processes. 146–147: 261–296. Bibcode:1995IJMSI.146..261G. doi:10.1016/0168-1176(95)04190-V.
6. ^ Marshall, Alan G.; Hendrickson, Christopher L.; Jackson, George S. (1998). "Fourier transform ion cyclotron resonance mass spectrometry: A primer". Mass Spectrometry Reviews. 17 (1): 1–35. Bibcode:1998MSRv...17....1M. doi:10.1002/(SICI)1098-2787(1998)17:1<1::AID-MAS1>3.0.CO;2-K. ISSN 0277-7037. PMID 9768511.
7. ^ Kanawati, B.; Wanczek, K. P. (2007). "Characterization of a new open cylindrical ion cyclotron resonance cell with unusual geometry". Review of Scientific Instruments. 78 (7): 074102–074102–8. Bibcode:2007RScI...78g4102K. doi:10.1063/1.2751100. PMID 17672776.
8. ^ Kanawati, B.; Wanczek, K. (2008). "Characterization of a new open cylindrical ICR cell for ion–ion collision studies☆". International Journal of Mass Spectrometry. 269 (1–2): 12–23. Bibcode:2008IJMSp.269...12K. doi:10.1016/j.ijms.2007.09.007.
9. ^ Cody, R. B.; Hein, R. E.; Goodman, S. D.; Marshall, Alan G. (1987). "Stored waveform inverse fourier transform excitation for obtaining increased parent ion selectivity in collisionally activated dissociation: Preliminary results". Rapid Communications in Mass Spectrometry. 1 (6): 99–102. Bibcode:1987RCMS....1...99C. doi:10.1002/rcm.1290010607.
10. ^ Shi, S; Drader, Jared J.; Freitas, Michael A.; Hendrickson, Christopher L.; Marshall, Alan G. (2000). "Comparison and interconversion of the two most common frequency-to-mass calibration functions for Fourier transform ion cyclotron resonance mass spectrometry". International Journal of Mass Spectrometry. 195–196: 591–598. Bibcode:2000IJMSp.195..591S. doi:10.1016/S1387-3806(99)00226-2.
11. ^ Leonardis, Irene; Chiaberge, Stefano; Fiorani, Tiziana; Spera, Silvia; Battistel, Ezio; Bosetti, Aldo; Cesti, Pietro; Reale, Samantha; De Angelis, Francesco (8 November 2012). "Characterization of Bio-oil from Hydrothermal Liquefaction of Organic Waste by NMR Spectroscopy and FTICR Mass Spectrometry". ChemSusChem. 6 (2): 160–167. doi:10.1002/cssc.201200314. PMID 23139164.
12. ^ Sudasinghe, Nilusha; Cort, John; Hallen, Richard; Olarte, Mariefel; Schmidt, Andrew; Schaub, Tanner (1 December 2014). "Hydrothermal liquefaction oil and hydrotreated product from pine feedstock characterized by heteronuclear two-dimensional NMR spectroscopy and FT-ICR mass spectrometry". Fuel. 137: 60–69. doi:10.1016/j.fuel.2014.07.069.
13. ^ Sleno L., Volmer D. A., Marshall A. G. (February 2005). "Assigning product ions from complex MS/MS spectra: the importance of mass uncertainty and resolving power". J. Am. Soc. Mass Spectrom. 16 (2): 183–98. doi:10.1016/j.jasms.2004.10.001. PMID 15694769.CS1 maint: Uses authors parameter (link)
14. ^ Bossio R. E., Marshall A. G. (April 2002). "Baseline resolution of isobaric phosphorylated and sulfated peptides and nucleotides by electrospray ionization FTICR ms: another step toward mass spectrometry-based proteomics". Anal. Chem. 74 (7): 1674–9. doi:10.1021/ac0108461. PMID 12033259.CS1 maint: Uses authors parameter (link)
15. ^ He F., Hendrickson C. L., Marshall A. G. (February 2001). "Baseline mass resolution of peptide isobars: a record for molecular mass resolution". Anal. Chem. 73 (3): 647–50. doi:10.1021/ac000973h. PMID 11217775.CS1 maint: Uses authors parameter (link)
16. ^ Solouki T., Marto J. A., White F. M., Guan S., Marshall A. G. (November 1995). "Attomole biomolecule mass analysis by matrix-assisted laser desorption/ionization Fourier transform ion cyclotron resonance". Anal. Chem. 67 (22): 4139–44. doi:10.1021/ac00118a017. PMID 8633766.CS1 maint: Uses authors parameter (link)
17. ^ a b Scigelova, M.; Hornshaw, M.; Giannakopulos, A.; Makarov, A. (2011). "Fourier Transform Mass Spectrometry". Molecular & Cellular Proteomics. 10 (7): M111.009431. doi:10.1074/mcp.M111.009431. ISSN 1535-9476. PMC 3134075. PMID 21742802.


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What is an adverb? An adverb describes a verb or an adjective or another adverb. It can describe when, where, or how about an action. When: After lunch, she he goes to sleep. Where: They went upstairs in search of the boys. How: Raja wrote a letter to his friend quickly. Note: Adverbs or adverb phrases can come at the beginning or end of a sentence. They can also be next to the verb or adjective they are describing.  Yesterday I went to a cinema. She likes to eat sweets on occasion. He desperately needed to get petrol for the car.
 What is an adverb? An adverb describes a verb or an adjective or another adverb. It can describe when, where, or how about an action. When: After lunch, she he goes to sleep. Where: They went upstairs in search of the boys. How: Raja wrote a letter to his friend quickly. Note: Adverbs or adverb phrases can come at the beginning or end of a sentence. They can also be next to the verb or adjective they are describing.  Yesterday I went to a cinema. She likes to eat sweets on occasion. He desperately needed to get petrol for the car.

format: plain


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The Warm Neptune GJ 3470b Has a Polar Orbit
Abstract

The warm Neptune GJ 3470b transits a nearby (d= 29 pc) bright slowly rotating M1.5-dwarf star. Using spectroscopic observations during two transits with the newly commissioned NEID spectrometer on the WIYN 3.5 m Telescope at Kitt Peak Observatory, we model the classical Rossiter–McLaughlin effect, yielding a sky-projected obliquity of$λ=98−12+15◦$and a$vsini=0.85−0.33+0.27kms−1$. Leveraging information about the rotation period and size of the host star, our analysis yields a true obliquity of$ψ=95−8+9◦$, revealing that GJ 3470b is on a polar orbit. Using radial velocities from HIRES, HARPS, and the Habitable-zone Planet Finder, we show that the data are compatible with a long-term radial velocity (RV) slope of$γ̇=−0.0022±0.0011ms−1day−1$over a baseline of 12.9 yr. If the RV slope is due to acceleration from another companion in the system, we show that such a companion is capable of explaining the polar and mildly eccentric orbit of GJ 3470b using two different secular excitation models. The existence of an outer companion can be further constrained with additional RV observations, Gaia astrometry, and future high-contrast imaging observations. Lastly, we show that tidal heating from more »

Authors:
; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; more »
Award ID(s):
Publication Date:
NSF-PAR ID:
10367476
Journal Name:
The Astrophysical Journal Letters
Volume:
931
Issue:
2
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Article No. L15
ISSN:
2041-8205
Publisher:
DOI PREFIX: 10.3847
National Science Foundation
##### More Like this
1. Abstract

Cosmic reionization was the last major phase transition of hydrogen from neutral to highly ionized in the intergalactic medium (IGM). Current observations show that the IGM is significantly neutral atz> 7 and largely ionized byz∼ 5.5. However, most methods to measure the IGM neutral fraction are highly model dependent and are limited to when the volume-averaged neutral fraction of the IGM is either relatively low ($x¯HI≲10−3$) or close to unity ($x¯HI∼1$). In particular, the neutral fraction evolution of the IGM at the critical redshift range ofz= 6–7 is poorly constrained. We present new constraints on$x¯HI$atz∼ 5.1–6.8 by analyzing deep optical spectra of 53 quasars at 5.73 <z< 7.09. We derive model-independent upper limits on the neutral hydrogen fraction based on the fraction of “dark” pixels identified in the Lyαand Lyβforests, without any assumptions on the IGM model or the intrinsic shape of the quasar continuum. They are the first model-independent constraints on the IGM neutral hydrogen fraction atz∼ 6.2–6.8 using quasar absorption measurements. Our results give upper limits of(1σ),$x¯HI(z=6.5)<0.87±0.03$(1σ), and$x¯HI(z=6.7)<0.94−0.09+0.06$(1σ). The dark pixel fractions atz> 6.1 are consistent with the redshift evolution of the neutral fraction of the IGM derived from Planck 2018.

2. Abstract

We report the discovery of MAGAZ3NE J095924+022537, a spectroscopically confirmed protocluster at$z=3.3665−0.0012+0.0009$around a spectroscopically confirmedUVJ-quiescent ultramassive galaxy (UMG;$M⋆=2.34−0.34+0.23×1011M⊙$) in the COSMOS UltraVISTA field. We present a total of 38 protocluster members (14 spectroscopic and 24 photometric), including the UMG. Notably, and in marked contrast to protoclusters previously reported at this epoch that have been found to contain predominantly star-forming members, we measure an elevated fraction of quiescent galaxies relative to the coeval field ($73.3−16.9+26.7%$versus$11.6−4.9+7.1%$for galaxies with stellar massM≥ 1011M). This high quenched fraction provides a striking and important counterexample to the seeming ubiquitousness of star-forming galaxies in protoclusters atz> 2 and suggests, rather, that protoclusters exist in a diversity of evolutionary states in the early universe. We discuss the possibility that we might be observing either “early mass quenching” or nonclassical “environmental quenching.” We also present the discovery of MAGAZ3NE J100028+023349, a second spectroscopically confirmed protocluster, at a very similar redshift of$z=3.3801−0.0281+0.0213$. We present a total of 20 protocluster members, 12 of which are photometric and eight spectroscopic including a poststarburst UMG ($M⋆=2.95−0.20+0.21×1011M⊙$). Protoclusters MAGAZ3NE J0959more »

3. Abstract

We present a stellar dynamical mass measurement of a newly detected supermassive black hole (SMBH) at the center of the fast-rotating, massive elliptical galaxy NGC 2693 as part of the MASSIVE survey. We combine high signal-to-noise ratio integral field spectroscopy (IFS) from the Gemini Multi-Object Spectrograph with wide-field data from the Mitchell Spectrograph at McDonald Observatory to extract and model stellar kinematics of NGC 2693 from the central ∼150 pc out to ∼2.5 effective radii. Observations from Hubble Space Telescope WFC3 are used to determine the stellar light distribution. We perform fully triaxial Schwarzschild orbit modeling using the latest TriOS code and a Bayesian search in 6D galaxy model parameter space to determine NGC 2693's SMBH mass (MBH), stellar mass-to-light ratio, dark matter content, and intrinsic shape. We find$MBH=1.7±0.4×109M⊙$and a triaxial intrinsic shape with axis ratiosp=b/a= 0.902 ± 0.009 and$q=c/a=0.721−0.010+0.011$, triaxiality parameterT= 0.39 ± 0.04. In comparison, the best-fit orbit model in the axisymmetric limit and (cylindrical) Jeans anisotropic model of NGC 2693 prefer$MBH=2.4±0.6×109M⊙$and$MBH=2.9±0.3×109M⊙$, respectively. Neither model can account for the non-axisymmetric stellar velocity features present inmore »

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5. Abstract

Benchmark brown dwarf companions with well-determined ages and model-independent masses are powerful tools to test substellar evolutionary models and probe the formation of giant planets and brown dwarfs. Here, we report the independent discovery of HIP 21152 B, the first imaged brown dwarf companion in the Hyades, and conduct a comprehensive orbital and atmospheric characterization of the system. HIP 21152 was targeted in an ongoing high-contrast imaging campaign of stars exhibiting proper-motion changes between Hipparcos and Gaia, and was also recently identified by Bonavita et al. (2022) and Kuzuhara et al. (2022). Our Keck/NIRC2 and SCExAO/CHARIS imaging of HIP 21152 revealed a comoving companion at a separation of 0.″37 (16 au). We perform a joint orbit fit of all available relative astrometry and radial velocities together with the Hipparcos-Gaia proper motions, yielding a dynamical mass of$24−4+6MJup$, which is 1–2σlower than evolutionary model predictions. Hybrid grids that include the evolution of cloud properties best reproduce the dynamical mass. We also identify a comoving wide-separation (1837″ or 7.9 × 104au) early-L dwarf with an inferred mass near the hydrogen-burning limit. Finally, we analyze the spectra and photometry of HIP 21152 B using the Saumon & Marley (2008)more »


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# Nonlinear Shrinkage

You are here:
• Nonlinear Shrinkage

### Nonlinear Shrinkage

Many statistical applications require an estimate of a covariance matrix and/or of its inverse when the matrix dimension, p, is large compared to the sample size, n.

A cursory glance at the Marˇcenko and Pastur (1967) equation shows that linear shrinkage is the first-order approximation to a fundamentally nonlinear problem. How good is this approximation? Ledoit and Wolf (2004) are very clear about this. Depending on the situation at hand, the improvement over the sample covariance matrix can either be gigantic or minuscule. When $$\frac{p}{n}$$ is large, and the population eigenvalues are close to one another, linear shrinkage captures most of the potential improvement over the sample covariance matrix. In the opposite case, that is, when $$\frac{p}{n}$$is small and the population eigenvalues are dispersed, linear shrinkage hardly improves at all over the sample covariance matrix.

### What can nonlinear shrinkage do?

The goal of solving covariance matrix is to find estimators that outperform the sample covariance matrix, both in finite samples and asymptotically. For the purposes of asymptotic analyses, to reflect the fact that p is large compared to n, one has to employ large-dimensional asymptotics where p is allowed to go to infinity together with n. And this is how nonlinear shrinkage works.

### Further Discussion

The estimation of covariance matrix can be divided into case $$pn$$

More details can be see at Spectrum estimation: A Unified Framework for Covariance Matrix Estimation and PCA in Large Dimensions.

### References

O. Ledoit, M. Wolf, Nonlinear shrinkage estimation of large-dimensional covariance matrices, Ann. Statist. 40 (2) (2012) 1024–1060.


text:
If α, β are the zeros of a polynomial such that α + β = −6 and αβ = −4,
Question:

If $\alpha, \beta$ are the zeros of a polynomial such that $\alpha+\beta=-6$ and $\alpha \beta=-4$, then write the polynomial.

Solution:

Let S and P denotes respectively the sum and product of the zeros of a polynomial

We are given $S=-6$ and $P=-4$. Then

The required polynomial $g(x)$ is given by

$g(x)=x^{2}-S x+P$

$g(x)=x^{2}-(-6) x+(-4)$

$=x^{2}+6 x-4$

Hence, the polynomial is $x^{2}+6 x-4$


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# Not allowed to suggest edits via SE app.

When I tried to edit a post via the SE android app, I got the red error message 'This account is not allowed to suggest edits'. I then tried the desktop website where it did work.

Is this a bug in the SE android app?

It is not possible to suggest edits through the app at present. This is a known limitation of Stack Exchange API, on which the mobile apps rely. See Android app should send edits to edit queue if necessary rather than rejecting them which has .

This part is outdated, per Dave Haney's comment

While it's true that bugs can be reported on local metas, for mobile apps it is much more efficient to report them on SE Meta. There's no point in scattering bug reports over 100+ per-site metas, when the subject matter has nothing to do with the sites themselves.

SE meta has the tag android-app which the app developers monitor closely. We don't, and even if we had one, it's not a given that developers will immediately find it and add it to their filter.

However, some points remain

• When posting on SE Meta, one is likely to find out (via suggested duplicates) that the bug was already reported, saving the trouble of writing a report and posting screenshots.
• Bug reports usually get a few upvotes on SE meta, and unlike on per-site metas, you get actual, shiny, precious reputation points there.
• I've added the first math meta android-app tag to this post. We have dev tools that let us look for the tag regardless of site, so it will appear for us even on math meta.
– Haney StaffMod
Jul 3 '14 at 18:14


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## Introduction

Cell-to-cell communication is a major driver of cell differentiation and physiological function governing organ development, homeostasis, and response to injury. Within tissues, cells have local neighbors with whom they directly communicate via paracrine signaling and direct cell–cell contact, and long-range or mobile partners with whom they exchange information via endocrine signaling. In solid tissues, cell types have specific cellular niches incorporating localized matrix and signaling environments, which facilitate phenotypic maintenance and support specialized cell functions. Circulating immune cells use an extensive library of chemokines to coordinate multicellular system responses to threat or injury. The advent of single-cell technologies has made it technologically possible, for the first time, to codify cell-specific ligand–receptor patterns in complex tissues with both high accuracy and robust statistical confidence. Numerous computational tools have emerged in the last years to mine intercellular communication information from single-cell data1,2,3,4,5,6,7,8. The combination of single-cell sequencing data with ligand–receptor mapping is therefore a promising approach to exploring, understanding, and reverse-engineering complex tissue systems-biology for biologic, therapeutic, and regenerative efforts. This manuscript formalizes and disseminates the techniques first published by Raredon et al.1, presenting open-source software to the wider community facilitating exact recapitulation of this analysis.

A connectomic network from a single tissue has unique properties that must be taken into consideration for biologically relevant downstream information processing. Tissue-derived connectomic networks are directional—i.e. each ligand–receptor interaction matrix is asymmetric; multi-modal—i.e. many ligand–receptor mechanisms contribute to the connectome; and weighted—i.e. interaction edges can be assigned quantitative values. These properties make data mining and data visualization substantially more complex than in some other genres of network science. Added dimensions can additionally come into play when it is necessary to compare cell–cell signaling in tissues between experimental conditions, over time during growth or remodeling, or between disparate tissue systems in which the same cell type annotations are not necessarily present. Here we describe a computational package in R called Connectome which facilitates each of these tasks. Connectome has been used to explore native signaling in human lung9 and to identify aberrant signaling in pulmonary arterial hypertension (PAH)10, chronic obstructive pulmonary disease (COPD)11, and COVID-1912.

Connectome is a multi-purpose tool designed to create ligand–receptor mappings in single-cell data, to identify non-random patterns representing signal, and to provide biologically-informative visualizations of these patterns. It can be applied to any single-cell dataset (or combination of datasets), and is designed to be used in association with the R package Seurat. By default, Connectome uses the FANTOM5 database of ligand–receptor interactions13, formalizing the workflow first presented in1, but it also allows mapping against any user-provided ligand–receptor list. Because the reference database can be customized, Connectome can also be used to investigate newly discovered or hypothesized ligand–receptor mechanisms of particular interest.

To demonstrate Connectome, we apply the software analysis in three distinct use-cases on public data. First, we demonstrate application to an individual tissue by analyzing single-cell human pancreas data. Second, we describe differential connectomics, comparing IFN-stimulated human PBMCs with unstimulated control data. Third, we apply Connectome to a longitudinal wound-healing dataset in mouse muscle. All software is available at https://github.com/msraredon/Connectome. Detailed vignettes and instructions for use are published at https://msraredon.github.io/Connectome/.

## Methods

### Definitions

The term connectome refers to the complete set of interactions between nodes in a cell system, in the same way that transcriptome refers to the complete set of transcribed genes within a cell. The term parcellation refers to the way that the above system is divided up into distinct nodes and in the application described in this manuscript is synonymous with celltype cluster. The way a system is parcellated strongly affects its node architecture and therefore the shape of the resulting connectome. An edge in Connectome is a single unique celltype–ligand–receptor–celltype interaction. Edge attributes are quantitative or qualitative pieces of information associated with an edge. A differential connectome is the network that results when two connectomes are directly compared, edge-for-edge. Centrality, as used in the text, refers to quantitative metrics of how ‘connected’ a given node is to other nodes, in either an outgoing (sending) or incoming (receiving) fashion.

### Defining the data structure of tissue connectomics

The connectomic mapping discussed here treats every cell parcellation (e.g., different cell types, phenotypically distinct cell states of the same cell type, etc.) as a single node, averaging ligand and receptor values across a given cell parcellation to yield mean values which are then linked. Mean-wise connectomics has the advantage of accommodating the zero-values intrinsic to single-cell data, while simplifying the system so that every cell parcellation is represented by a single, canonical node. However, as this approach will blend the effects of all cellular archetypes within a cluster, initial cell parcellation must be done carefully for the resulting connectomic networks to be biologically meaningful.

Connectome, by default, calculates two distinct edgeweights, each of which captures biologically relevant information. The first edgeweight, also referred to as ‘w1’ or ‘weightnorm,’ is defined as the product of the celltype-wise normalized expression for the ligand and the receptor, or

$$w_{1}^{ijk} = weight_{norm}^{ijk} = product \left(Celltype_{i}^{{L_{k}^{normalized} }} ,Celltype_{j}^{{R_{k}^{normalized} }} \right)$$

where (k) denotes a specific ligand–receptor pair, $$Celltype_{i}^{{L_{k}^{normalized} }}$$ is the normalized expression of ligand $$L_{k}$$ in celltype (i), and $$Celltype_{j}^{{R_{k}^{normalized} }}$$ is the normalized expression of receptor $$R_{k}$$ in celltype (j). Weightnorm is useful for differential connectomics, in which exact edges are compared across tissue conditions. However, Weightnorm is encumbered by the fact that it ignores the cell-type specificity of many ligands and receptors. Weightnorm will weight a highly expressed but non-specific cell–cell link as stronger than a highly specific but lowly-expressed gene. This does not align with biologic intuition for many cell–cell signaling mechanisms, nor with the goal of parsing out rare cell types that may have outsized biological relevance. Therefore, Connectome also calculates a second edgeweight, alternatively referred to as ‘w2’ or ‘weightscale,’ which is defined as the mean of the gene-wise z-scores of the ligand and receptor, or

$$w_{2}^{ijk} = weight_{scale}^{ijk} = mean \left(Celltype_{i}^{{L_{k}^{scaled} }} ,\; Celltype_{j}^{{R_{k}^{scaled} }} \right)$$

where $$Celltype_{i}^{{L_{k}^{scaled} }}$$ is the system-scaled expression of ligand $$L_{k}$$ in a celltype (i) and $$Celltype_{j}^{{R_{k}^{scaled} }}$$ is the system-scaled expression of receptor $$R_{k}$$ in a celltype (j). Because weightscale incorporates information on the topology of the entire system being analyzed, leveraging system z-scores of each gene, it is ideally suited to exploring cellular ‘roles’ within a single tissue system. It also can be used to compare cellular prominence within signaling networks across disparate tissue systems, species, or batches, in which normalized values may vary but their relative expressions across cell types do not1.

Alternative edgeweights, such as those incorporating a score for likelihood of downstream signal transduction5 or those which take into account proximity information from spatial transcriptomics or histology-registered techniques14, may reveal additional patterns. For the purposes of this software program, Connectome defaults to the above edgeweights definitions because we have found them to be effective for understanding biological questions addressed in our studies.

### Identification of edges of interest or significance

Statistical significance depends upon the question being asked, the test being applied, and the threshold for important differences, and has a very different meaning depending on whether a research team is investigating a single tissue, comparing multiple tissues, or studying a physiologic process. We find two general statistical patterns to be of key importance, and we have built-in functionality to Connectome to help the researcher focus on these patterns.

#### Single-tissue system

First, when studying a single tissue system (i.e., one organ in one condition), it is desirable to focus on those ligand–receptor interactions which are highly associated specific celltype-celltype pairings. To do so, we recommend limiting analysis to only those edges where the ligand and the receptor are both expressed in greater than a certain fraction of their respective clusters, generally 10%, and to then further limit analysis to those edges where both the ligand and the receptor have a p-value of less than a given threshold as determined by a system-wide Wilcoxon Rank Sum Test (calculated by default within CreateConnectome.) These thresholds can be further reduced within FilterConnectome, to identify those pathways of greatest significance, and can be crossed with additional thresholding to limit edge representation to specific vectors or signaling families (see vignettes online).

#### Two-tissue comparison

When comparing two tissue systems (i.e., one organ in two experimental conditions) it is generally interesting to focus on those mechanisms which, regardless of their association with a specific celltype-celltype vector, are differentially regulated across condition.

The vignette online shows how to use a Wilcoxon Rank Sum test to compare each pair of commensurate cells across two datasets and to thereby determine which ligands and which receptors are differentially expressed to a statistically significant degree. The differential connectome can then be thresholded to only those edges where both the ligand on the sending cell and the receptor on the receiving cell are differentially expressed between experimental conditions. All differential connectomics presented and discussed in this manuscript and online utilize this statistical approach.

### Centrality definition and usage

Centrality and CompareCentrality both calculate and plot-to-compare two key centrality metrics for networks of interest: the cumulative incoming/outgoing edgeweight for each node, and the Kleinberg Hub and Authority scores for each node. These two centrality metrics often correlate, but they do not always agree on node ranking. There is no single, proper and agreed upon way to best calculate network centrality16; these two metrics were chosen because they are commonly used in network research and are biologically interpretable. Cumulative incoming/outgoing edgeweight fraction is the sum of the incoming and outgoing edges, calculated per node, expressed by default as the fraction of total edgeweight within the considered system; this ranks each node’s ability to contribute or listen to a given set of interactions. Hub-Authority centrality is a heavily-used information flow metric17, which should be interpreted, in this instance, to reveal Hubs which send signal to Authorities, and Authorities which receive information from Hubs. Each of these metrics provide interpretable information regarding biologic network structure.

### Perturbation score definition and usage

In a differential connectome, every edge may fall into one of four distinct categories, ligand UP/receptor UP, ligand DOWN/receptor DOWN, ligand UP/receptor DOWN, or ligand DOWN/receptor UP. If both the ligand and receptor are upregulated, we may reasonably call the edge ‘activated’, while conversely, when both the ligand and receptor are downregulated, we may think of the edge as ‘deactivated’. The two alternate cases, when a ligand is upregulated and a receptor down, or vice-versa, are more complicated to interpret. Although such patterns may initially suggest ligand–pressure or ligand–starvation, in which the downstream cell changes its character in response to upstream influence, it is important to note that in most tissue systems there are many cells interacting at once, and that there are likely multi-cell feedback loops in play which confound easy interpretation of such patterns.

To accommodate this data architecture and identify strongly perturbed edges regardless of category, we defined a ‘perturbation score’ that is calculatable for each edge, expressed as the product of the absolute values of the log-fold change for both the receptor and ligand, or:

$$score^{ijk} = \left| {log_{2} \left( {\frac{{Celltype_{i}^{{L_{k}^{test} }} }}{{Celltype_{i}^{{L_{k}^{control} }} }}} \right)} \right| \times \left| {log_{2} \left( {\frac{{Celltype_{j}^{{R_{k}^{test} }} }}{{Celltype_{j}^{{R_{k}^{control} }} }}} \right)} \right|$$

where $$Celltype_{i}^{{L_{k}^{test} }}$$ and $$Celltype_{i}^{{L_{k}^{control} }}$$ are the normalized expression of the ligand in celltype (i) in the test and control condition, respectively, and $$Celltype_{j}^{{R_{k}^{test} }}$$ and $$Celltype_{j}^{{R_{k}^{control} }}$$ are the normalized expression of the receptor in celltype (j) in the test and control condition, respectively. The perturbation score is calculated in the above fashion so that edges with large changes on both the ligand and receptor side, regardless of sign, have very high scores, and minimally perturbed edges have a score of zero. The perturbation score therefore varies from 0 to Infinity, and increases proportionally to both the log fold-change of the ligand (in either direction) and to the log fold-change of the receptor (in either direction). It is generally useful to limit analysis to those edges which have ligand and receptor expression in > 10% of their respective clusters in either the control or the test condition to avoid noisy measurements; this thresholding functionality is built into CircosDiff and DifferentialScoringPlot.

### Signaling family categorization

The signaling family categorizations used in this manuscript are carried over from1 with minor updates per recent literature findings. These groupings are loaded by default within Connectome. It should be noted that each signaling mechanism is allowed only a single designation in this formulation; in biologic reality, however, many signaling mechanisms can be considered to belong to multiple signaling families. These groupings are meant as a guide for hypothesis generation and later downstream exploration, rather than as definitive classification.

### Analysis of pancreas data

In brief, the panc8 data was downloaded from SeuratData, normalized, scaled, and run through the CreateConnectome with a min.cells.per.ident cutoff of 75. Downstream analysis was limited to those edges where both the ligand and the receptor were expressed in > 10% of their respective clusters, and which had a p-value < 0.05 for both the ligand and the receptor. Additional thresholds applied for specific visualization purposes are emphasized when appropriate in the figure legends.

For the centrality analysis present in Fig. 2, Centrality was first run across all signaling families. Each signaling family was then grouped according to which of the 4 cell classes was found to dominate incoming centrality within the given dataset. Centrality was then re-run 4 times, once for each set of signaling families, to yield the 4 sub-graphs.

### Analysis of IFN-stimulated vs. Control PBMC data

In brief, the ‘ifnb’ dataset was loaded from SeuratData. Each dataset was normalized, scaled, and run through CreateConnectome. The two connectomes were then passed to DifferentialConnectome. A Wilcoxon Rank Sum test was then performed for each individual cell across conditions, for all ligands and all receptors, and the results from this test were used to limit the differential connectome to only those edges showing statistically significant differences (p < 0.05) for both the ligand and the receptor. Downstream analysis was further limited to those edges which had ligand and receptor expression in > 10% of their respective clusters in either the control or the test condition. Figure 3C was made with DifferentialScoringPlot and Fig. 3D was made with CircosDiff, both of which are built in to Connectome.

### Analysis of muscle wound-healing data

In brief, raw data was loaded from De Micheli et al.18, normalized, scaled, and run through CreateConnectome. ggplot was use for Fig. 4A,D, CircosPlot for Fig. 4B, and CompareCentrality for Fig. 4C.

## Results

### Visualizing the connectomic signature of a single tissue

Figure 1 shows the workflow for ligand–receptor connectomics in a single tissue. In this instance we use single-cell data from 8 separate single-cell RNA libraries of human pancreatic tissue, available through the SeuratData database19. As a first step (Fig. 1A), the tissue data are parcellated into defined cell types using a standard clustering workflow. Then, a mapping is created against a ground-truth database of known ligand–receptor mechanisms. This mapping yields a large edgelist, wherein nodes are defined as cell types and edge attributes contain quantitative information derived from cell type-specific expression levels. Source nodes denote the ligands, and target nodes refer to the cognate receptors. This data architecture is stored as a data frame in R, the environment which is particularly amenable to subsetting networks of interest and working with graph theory-based computational packages. This process is performed by a single function in Connectome which allows customization of edgeweight definition and edge attribute calculations. By default, two edgeweights are calculated: weightnorm and weightscale, both of which are defined in “Methods”. Weightnorm is a measure of raw connectivity between two nodes, while weightscale is a measure of signal specificity between two nodes within a given tissue system. Both metrics are biologically informative.

Conceptually, the data architecture for a single edge attribute (one column in the above discussed edgelist data frame in Fig. 1A) can be thought of as a 3D matrix (Fig. 1B), where rows are source (sending) cells, columns are target (receiving) cells, and the z-axis is the full list of ground-truth known ligand–receptor mechanisms. This allows clear visualization of the data that needs to be subset in order to explore a single interactome (red), outgoing network (purple), niche network (blue), or cell–cell vector (green). The blue rectangle represents a niche-network, containing information on all edges in a position to influence a single cell type. The purple rectangle shows all edges coming from a single cell type. The red rectangle is a single interactome, containing information for a single signaling modality between all cell types. The green prism represents a single cell-to-cell vector, containing information for all signaling modalities.

Visualization of these connectomic networks can be done in multiple ways. Connectome includes a series of functions designed for tissue network exploration, including one which generates plots allowing immediate quantitative visualization of individual interactomes (e.g. Fig. 1C), celltype-to-celltype vectors (Fig. 1D) and niche networks (Fig. 1E). Further, the similarity between individual celltype-to-celltype vectors (vectortypes) can be analyzed using a k-nearest-neighbor style embedding (Fig. S1). This style of visualization places vectortypes in a 2-dimensional space and provides a quantitative way to cluster vectortypes based on which ligand–receptor mechanisms are most highly weighted in each celltype-to-celltype pairing. We provide a custom function in Connectome to perform this analysis.

### Tissue network centrality analysis

A major goal of tissue science and cell biology is to understand the roles that individual cells types play and their ability to affect other cell types. It is of interest, therefore, to rank cell types based on their ability to produce or receive information within a given signaling network, or signaling family. To quantify these roles, Connectome is capable of performing a centrality analysis, an example which is shown in Fig. 2. In this analysis, the total connectome for a single tissue is first subset down to only those edges belonging to a single signaling family (Fig. 2A). This weighted graph is then used to calculate two centrality metrics: the Kleinberg Hub and Kleinberg Authority scores (represented as the dot size), and the cumulative directed edgeweight (x-axis), for each cell type within the network. A high Hub score, biologically, implies that a sending cell type is producing high levels of ligand which are sensed by other cells in the system. A high Authority score, biologically, means that a receiving cell type is highly expressing receptors capable of sensing ligands secreted by other cells in the system. The dot sizes arising from the Kleinberg scores are then used to visualize outgoing and incoming centrality of each cell type (Fig. 2B), for a variety of specific signaling mechanisms. For example, the endothelial cell population of the pancreas has a high outgoing centrality score for signaling along the PDGB axis to the mesenchyme compartment (green circle in upper left panel of Fig. 2B, “endothelial”, highly connected to the ochre circle in upper right panel, ‘activated stellate’). The result is a ranking of outgoing and incoming centrality for every cell type, across every signaling family, for each target cell type within the tissue. Collectively, this analysis creates a comprehensive portrait of potential extracellular signal transfer within a given tissue system. As shown in Fig. 2, this information can then be used to group each signaling family based on the cell type that is best positioned to receive information (shown, sorted by incoming centrality) or best positioned to generate a signal (not shown—requires sorting by outgoing centrality).

### Differential connectomics with aligned nodes

It is common in systems biology to examine changes in cell–cell signaling between two systems, i.e., in a multicellular tissue before and after treatment with a drug or chemokine (Fig. 3A). In one such example, if the same cell types are present in both systems, we may calculate direct, one-to-one comparison of all edges using the Connectome package. It should be noted, however, that within a differential connectome, there are four distinct types of perturbed edges, since for each cell in a differential edge, the ligand and the receptor may be either increased or decreased (see “Methods”, Fig. 3B). Figure 3 shows the utility of this concept in the interferon-simulated vs. control dataset of human peripheral blood mononuclear cells available through SeuratData20. Figure 3C shows log fold changes for selected ligands and receptors, and the resulting perturbation scores (which are always positive) for each cell–ligand–receptor–cell edge. Perturbed edges can then be grouped by their differential pattern and their perturbation scores visualized in readily-explorable network form (Fig. 3D).

### Longitudinal connectomics for a dynamic tissue-system process

Comparing tissue-level connectomics datasets can be difficult when the cell nodes are not directly comparable between the two tissue systems. Such a situation can occur if new cell types are recruited into, or eliminated from, a tissue system. Alternatively, normal differentiation processes may take place which cause a new cell type to emerge. Such changes occur frequently, whether due to inflammation, response to injury, wound healing, or normal development. Many unanswered questions in tissue science center around how these changing cellular landscapes correlate with shifting cell–cell communication patterns that are present in tissues and organs. Because centrality analysis quantifies network topology while being agnostic to specific nodes being present, the described technique allows for the comparison of disparate tissue systems which do not necessarily contain the same cell types.

As an example of a comparison of tissue systems containing different cell types, the muscle wound-healing dataset recently published by De Micheli et al.18 provides an excellent use-case for longitudinal tissue connectomics. In this study, done in vivo in mice, muscle tissue was injured and then allowed to heal over time. scRNAseq was performed on Day 0, immediately before injury, and then on Days 2, 5, and 7 post-injury. Cells were parcellated on a per-time-point basis, leading to clear trends in cell type tissue fractions over time (Fig. 4A). Certain cell types are present in the tissue throughout the wound healing process [i.e., endothelial cells, fibro/adipogenic progenitors (FAPs)], while others are recruited into the tissue solely during acute wound healing (i.e., monocytes/macrophages/platelets).

For demonstration, we explore the network topology of cell–cell signaling based on Vascular Endothelial Growth Factor A (Vegfa) over time. Figure 4B–D suggests that there is a dramatic change in the dominant source of this ligand during wound healing. In the baseline state, FAPS are the dominant producers of Vegfa. Immediately after injury, FAPs reduce their production of this ligand, and newly recruited Monocytes/Macrophages/Platelets take up dominant production of Vegfa. On Day 5, these two cell types share this functional role, and at the conclusion of healing, FAPs again dominate the network. Endothelial cells, meanwhile, are always expressing a panoply of receptors for this secreted factor and are in a prime position to receive angiogenic information. We see that supporting cell types, including smooth muscle cells and tenocytes, are also in a position to receive Vegfa-mediated information before injury and after the conclusion of healing (green sectors). Further, we observe that muscle stem cells (MuSCs) are also capable of sensing aspects of Vegfa-mediated signaling, in particular on Day 5 post-injury, when they co-express and upregulate established angiogenesis-modulating receptors Egfr, Gpc1, and Itgb121,22,23. It should be noted that, because of the parcellations chosen, this technique cannot necessarily tell the difference between an entire population of cells shifting in character versus a new phenotypic archetype emerging within an existing population. Further sub-clustering (i.e. finer, follow-up parcellation) is currently required to explore these kinds of questions.

## Discussion

Connectome is a multi-purpose toolset which can be used to map, explore, and visualize patterns of ligand–receptor expression in any single-cell dataset. It generalizes the workflow first developed in1 and allows adoption by the wider community. The software is open-source on GitHub and includes vignettes to replicate and adapt all analyses discussed. It allows quantification and observation of both fine-grain (single-mechanism, single-celltype-to-celltype vector), and coarse-grain (total signaling family, cross-system) connectivity patterns.

There are a number of clear caveats to the above described techniques. First and foremost, strongly paired ligand and receptor expression is not direct evidence of cell–cell communication. For many ligand–receptor mechanisms, cells must be in direct or very close proximity in order to communicate via that mechanism. Suspension-based single-cell sequencing, however, does not preserve spatial histologic information, and so the techniques described here attempt to make a qualitative portrait of a tissue based on what interactions are possible; any robust claim of transduction requires extensive wet-lab experimentation. Although intracellular transduction can be computationally predicted5, the complexity of intracellular signaling networks makes this task fundamentally confounded for many ligand–receptor mechanisms.

Secondly, the outputs of Connectome, like any ligand–receptor mapping software, are only as good as the ground-truth database against which the original mapping takes place. This is why we have made the ground-truth customizable in this platform. Customization allows removal of extraneous connections with low biologic relevance, or addition of newly-discovered or researcher-hypothesized mechanisms. The software here utilizes the FANTOM5 database without modification, but iterative application has shown that a custom database can be useful for many specific researcher inquiries, in particular in immunology where complex immune cues are of interest.

Connectome is designed to be a fundamental tool for single cell researchers, computational biologists, and tissue engineers. It is intended to allow rapid, low-computationally-intensive-access to cell–cell signaling patterns that are present in single-cell data. Our intent is to allow researchers to quickly identify strongly-expressed signaling genes, to find strong pairings between cell types within identified signaling mechanisms, and to condense large amounts of network-level connectivity information into simple, quantitative plots which reflect the structure of tissue systems. We show here that Connectome can be applied to individual tissues, paired experimental conditions, and longitudinal datasets. In each case, Connectome yields biologically relevant information that can be used to help answer, and inform, specific questions regarding biological systems.


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MathOverflow will be down for maintenance for approximately 3 hours, starting Monday evening (06/24/2013) at approximately 9:00 PM Eastern time (UTC-4).

I'm having problems understanding why the conjugate gradient method breaks down for singular matrices. I've read a good introduction to intuitively understanding the CG method through visualizing the quadratic form, but I'm not really understanding mathematically why this seems to be. Does anyone have a good mathematical explanation or any sources that explain why conjugate gradient iteration doesn't work for singular matrices?

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 In what sense would you want it to work? Typically, a system won't have any solution at all if the coefficient matrix is singular. – Darsh Ranjan Dec 5 2009 at 22:04 My question was geared more towards any mathematical explanation of why singular matrices don't work. My understanding, from the previously referenced article (pg 5), is that singular matrices have multiple solutions rather than a single solution. – john Dec 5 2009 at 22:30

If the matrix is singular then then the images of some vectors may be zero and that could mean some of the steps may end up with division by zero.

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My question was geared more towards any mathematical explanation of why singular matrices don't work. My understanding, from the previously referenced article (pg 5), is that singular matrices have multiple solutions rather than a single solution.

Unfortunately, that's not so. If $A$ is a singular matrix, then for most $b$, the equation $Ax=b$ has no solution. It has a solution if and only if $b$ is in the column space of $A$ (or orthogonal to the kernel of $A$, if $A$ is symmetric), in which case there are infinitely many solutions. This is basic linear algebra.

Let's say $A$ is symmetric positive semidefinite, so we can at least entertain the thought of conjugate gradients. Conjugate gradients can be interpreted as maximizing the quadratic function $$f(x) = b^Tx-\frac12 x^TAx.$$ If $A$ is singular, then this function typically has no maximum value, as I will demonstrate. Since $A$ is singular, there are vectors $v$ such that $Av = 0$. If there is any such vector $v$ that is not orthogonal to $b$, consider substituting $x=tv$ into $f(x)$: $$f(tv) = t(b^Tv) - \frac12t^2(v^TAv) = t(b^Tv) = ct,$$ where $c$ is the nonzero real number $b^Tv$. Obviously, this is unbounded as a function of $t$, so we see that there is no maximum value if $b$ is not orthogonal to the kernel of $A$.

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The conjugate gradient method becomes unstable when the matrix A is singular, i.e., if you compute the output of the k-th iteration for a small change in the initial data, large deviations will occur. A similar large deviation occurs due the computer roundoff error. Even for non singular matrices A that are close to to be singular, for example when one of the eigenvalues of A is smaller by orders of magnitudes than the rest, instability problems can occur: The convergence rate becomes slower, and even convergence can be lost due to roundoff errors.

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A search for "conjugate gradient singular matrix" took me to this question. While the answer is obviously given by the responses, the question can be refined: Can CG still give a working algorithm if the matrix is singular, but behaves as a symmetric positive definite form on a (large) subspace?

A standard example is given by the finite element discretization of the Neumann problem on a simply connected domain. The constant functions are both the kernel and the cokernel of the Laplacian. On functions with vanishing mean, the Laplacian is still a positive definite symmetric operator, and we would like to leverage this structure.

This is non-trivial and best our numerical method is derived from a fully analytic setting, because this might provide us the convergence analysis as well. --- This appraoch is for example elaborated in

On the Finite Element Solution of the Pure Neumann Problem
Pavel Bochev and R. B. Lehoucq
SIAM Review
Vol. 47, No. 1 (Mar., 2005), pp. 50-66


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Thread: Show that f is convex on an interval

1. Show that f is convex on an interval

Hi there,
this is one of those problems I can't even figure out how to approach. Any idea would be highly appreciated.

Prove that
f is convex on an interval iff for all x and y in the interval,
f(tx + (1 t)y) < tf(x) + (1 t)f(y), for 0 < t < 1

2. Originally Posted by dgmath
Hi there,
this is one of those problems I can't even figure out how to approach. Any idea would be highly appreciated.

Prove that
f is convex on an interval iff for all x and y in the interval,
f(tx + (1 t)y) < tf(x) + (1 t)f(y), for 0 < t < 1

As far as I am concerned this is exactly the definition of (upwards) convexity...which definition do you have?

Tonio

3. Originally Posted by tonio
As far as I am concerned this is exactly the definition of (upwards) convexity...which definition do you have?

Tonio
[/left]
I almost guarantee based on dgmath's previous posts that his definition is that $f''(x)>0$. Of course, he/she missing some extra stipulations...differentiability etc.

4. Hi there,
I guess for the same reason, I'm stuck here. what I posted is pretty much all I have, but it IS true that it is a rephrased version of definition of convexity.


text:
Mathematics is the music of reason ~
James Joseph Sylvester

In my last post in this series on the Witten-equation, I showed that the genus-zero Landau-Ginzburg/Calabi-Yau correspondence amounts to the assertion that there is a degree-preserving isomorphism between the two state spaces and that after certain identifications, the small $J$-functions coincide, thus, the 'Picard-Lefschetz' Witten relation

'Edward Witten is often likened to Einstein; one colleague reached even further back for a comparison, suggesting that Witten possessed the greatest mathematical mind since Newton' ~ John Horgan

This is a series of posts that hopefully will culminate in a book I am working on. The aim is to show that the Witten equation in the context of Picard-Lefschetz Theory, leads to the non-forking, super-stability and hyper-categoricity of M-theory, thus making it the 'only-game-in-town' indeed. First, I will analyse the Witten nonlinear elliptic system of PDEs associated with a quasi-homogeneous polynomial-super-potential by showing its depth via the Landau-Ginzburg/Calabi-Yau correspondence.

Ex nihilo nihil fit ~ Parmenides

In this post, in order to solve the Wheeler-deWitt problem-of-time discussed by me, and since string/M-theoretic braneworld cosmology offers the only solution in a unified C*-Heisenberg-algebraic sense, I must first analyse a field theory on a D-brane sigma worldvolume and study the Dirac-Born-Infeld action


text:
# Need help with find the cartesian equation of a parametric set.

#### Psykik

[FONT=&quot]I have a Lissajous curve with the equations x=cos(2*t) and y=sin(t) for 0<t<2(pi). I need to find thae cartesian equation for this and I am really stuck. Any help is greatly appreciated[/FONT]

#### Idea

trig identity

$$\cos (2t)=1-2 \sin ^2t$$

1 person


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# How to find a simultaneous solution to these equations?

QUESTION

$$x^2 + xy + yz + xz = 30$$

$$y^2 + xy + yz + xz = 15$$

$$z^2 + xy + yz + xz = 18$$

I have tried manipulating the expressions, the identity of $$(x+y+z)^2$$ but to no avail.

Along with the answer, it would be great if you can explain the approach and thought process involved in dealing with such kind of questions.

Hint: Let $$x+y = a, y+z = b, z + x = c$$.
Rewrite the equations in term of $$a,b,c$$.

The approach and thought process is by recognizing the pattern of how the LHS factorizes.

• I believe that would be z+x=c. – InfiniteCool23 Oct 24 at 6:39
• Anyway, thanks for your help! – InfiniteCool23 Oct 24 at 6:40
• It works to give x=4,y=1,z=2 – InfiniteCool23 Oct 24 at 6:41

You didn't say whether $$x,y,z$$ are real or integer. But let's see what happens if we assume they're integer.

We can eliminate the cross terms $$xy + yz + xz$$ by subtraction.

\begin{align} x^2-y^2 &= 15\\ x^2-z^2 &= 12\\ z^2-y^2 &= 3\\ \end{align}

Now factorizing,

$$z^2-y^2 = (z+y)(z-y) = 3$$ So (if they're positive integers), $$z+y=3 \, \text{and} \, z-y=1$$ thus $$z=2, y=1$$. If we permit negative integers, $$z=-2, y=-1$$ also work.

Now substituting, $$x^2-4=12$$ so $$x=\pm 4$$.

And these values are consistent with the first equation: $$4^2-1^2=15$$ so we are done.


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# In the standard form of the equation for a parabola, what is represented by a?

Jan 21, 2015

I'm assuming that by standard form, you mean that the parabola is expressed with the equation $y = a {x}^{2} + b x + c$. So, $a$ is the quadratic coefficient.

First of all, it is a general rule that, given the plot of a function $f \left(x\right)$, the plot of $f \left(x\right) + c$ will simply be translated vertically, upward if $c$ is positive, or downward if $c$ is negative. So this is the role of the coefficient $c$.

As for the coefficient $a$ and $b$, they give us information about the shape of the parabola and its position on the plane.

In fact, we know that the abscissa of the vertix is given by $- \setminus \frac{b}{2 a}$, so both $a$ and $b$ coefficients tell us where the vertix is.

Moreover, we know that the behaviour at infinity of a polynomial is given by the highest power of $x$; and parabola is nothing but a polynomial of degree 2. So, the limit at both $\setminus \pm \setminus \infty$ are both the same infinite, and the $a$ coefficient tells us its positivity: if $a$ is positive, we have that
$\setminus {\lim}_{x \setminus \to \setminus \pm \setminus \infty} a {x}^{2} + b x + c = \setminus \infty$
while if $a$ is negative,
$\setminus {\lim}_{x \setminus \to \setminus \pm \setminus \infty} a {x}^{2} + b x + c = - \setminus \infty$

As an example, you can confront the plot of the simplest parabolas with $a > 0$ or $a < 0$, i.e. $y = {x}^{2}$ ($a = 1$) and $y = - {x}^{2}$ ($a = - 1$). Here's their graphs: you can see that the first one faces upwards, the second downward.

$y = {x}^{2}$
graph{x^2 [-4.77, 5.23, -0.897, 4.313]}

$y = - {x}^{2}$
graph{-x^2 [-5.02, 4.98, -4.94, 0.27]}


text: In other cases, you may derive the function from repeated experiments or observations. ... From the value of torque and inertia,the angular acceleration is found using equation, Procedure for doing Real Lab . in rad/s2 for this part i got 46.07 but its incorrect, i used the formula and even converted the rpm's to rad.per sec but still got it wrong. Converting to radians/s gives: ω = 0.125 rev/s * 2π rad/rev = 0.785 rad/s . Books. f the CD rotates at 500 (revolutions per minute) while the last song is playing, and then spins down to zero angular speed in 2.60 with constant angular acceleration, what is , the magnitude of the angular acceleration of the CD, as it spins to a stop? This article was co-authored by our trained team of editors and researchers who validated it for accuracy and comprehensiveness. Angular position is usually expressed in radians. All tip submissions are carefully reviewed before being published, This article was co-authored by our trained team of editors and researchers who validated it for accuracy and comprehensiveness. Positive motion is measured in a counterclockwise direction. To convert the number of radians to the number of revolutions, recall that 1 full circle (or 1 revolution) is equal to 2pi radians. For the example of the CD player, the calculation is as follows: For the roller coaster example, the calculation looks like this: Note that acceleration is always going to be in units of some distance measurement “per” time squared. This is nothing but the counterpart of s = 1/2 * (v0 + v1)t = v0*t + 1/2 a*t^2 in the rotational world. Then, find the derivative of the function for angular velocity in order to determine the function for angular acceleration. =1/2 x ( 104 rad/s 2 * (3.5s) 2 ) θ = 637 rad. … References. Revolutions per minute (abbreviated rpm, RPM, rev/min, r/min, or with the notation min −1) is the number of turns in one minute. Please consider making a contribution to wikiHow today. Because 1 rev=2π rad, we can find the number of revolutions by finding θ in radians. If it is slowing down, the acceleration is negative. Somehow, an ant is stuck to the rim of a bicycle wheel of diameter 1 m. While the bicycle is on a central stand, the wheel is set into rotation and it attains the frequency of 2 rev/s in 10 seconds, with uniform angular acceleration. The roller coaster, after applying its brakes to the spinning wheels, ultimately reaches an angular velocity of zero when it stops. This implies that; N = Number of revolutions per minute = 60. ω = 2πN / 60 ω = 2 x π x 24 / 60 ω = 150.816 / 60 ω = 2.5136 Hint 1. n= 100 revolutions. If you were further told that the light beam has a range of 10 meters, what would be the tip of the beam's linear velocity v, its angular acceleration α and its centripetal acceleration ac? For example, if a motorcycle wheel has a large angular acceleration for a fairly long time, it ends up spinning rapidly and rotates through many revolutions. ... You are on a ferris wheel that is rotating at the rate of 1 revolution every 8 seconds. To measure negative acceleration over a braking distance, you can use this measurement as the initial velocity. This article shows how to find acceleration in radians per second squared. The acceleration of gravity is 9.8 m/s 2. Alternatively, you may have a function to calculate the object's position. By signing up you are agreeing to receive emails according to our privacy policy. Next, find the angular velocity, which is the measure of how fast the object changes its position. How many revolutions will it make during this time ? An automobile engine slows down from 3600 rpm to 800 rpm in 6.0 s. Calculate the total number of revolutions the engine makes in this time. 10.4k SHARES. For this article, we assume that the function has been provided or previously calculated. Since this is a constant angular acceleration, you can easily calculate the average angular velocity by (w0 + w1) / 2. ... One way to find the number of revolutions the wheel undergoes as it slows to a stop is to … wikiHow is where trusted research and expert knowledge come together. Compute the angular acceleration and the number of revolutions made in… wikiHow's Content Management Team carefully monitors the work from our editorial staff to ensure that each article is backed by trusted research and meets our high quality standards. Amid the current public health and economic crises, when the world is shifting dramatically and we are all learning and adapting to changes in daily life, people need wikiHow more than ever. One full rotation around the unit circle is said to measure 2π radians. In more technical terms, if the wheel’s angular acceleration $$\alpha$$ is large for a long period of time $$t$$ then the final angular velocity $$\omega$$ and angle of rotation $$\theta$$ are large. An electric fan is turned off, and its angular velocity decreases uniformly from 500 rev/min to 200 rev/min in 4.00 s. (a) Find the angular acceleration in rev/s$^2$ and the number of revolutions made by the motor in the 4.00-s interval. Sometimes it is useful to convert from radians to degrees. you spin a wheel twice, then its total angle rotated is 4 pi, or 2 * 2 pi [since 2 pi is one full revolution]). The angular acceleration formula is derived in the same essential way as the angular velocity … Thanks to all authors for creating a page that has been read 66,198 times. In some cases, you may be provided with a function or formula that predicts or assigns the position of an object with respect to time. To calculate instantaneous angular acceleration, start by determining the function for angular position, or the position of the object with respect to time. After 60 more complete revolutions, its angular speed is 12.389 rev/s.. Calculate (i) Number of revolutions completed by the ant in these 10 seconds. From observations of roller coasters being tested, it has been found that they can come to a complete stop within 2.2 seconds from when the brakes are initially applied. ANSWER: Correct … Angular acceleration: α = a t /r Note that the angular acceleration is connected to the tangential acceleration, not the centripetal acceleration. The sign is conventionally taken to be positive if the radius vector turns counter-clockwise, and negative if clockwise. This problem has been solved! As an alternative example, suppose you know from test measurements that the wheels of a roller coaster spin at a velocity of 400 revolutions per second, which is equivalent to 2513 radians per second. wikiHow's. Angular acceleration is expressed in units of radians per time squared. A pulley 1 metre in diameter … A compact disc plays in the machine by rotating at an angular velocity of 160 radians per second. The crankshaft in a race car goes from rest to 3500 rpm in 3.5 seconds. If you know the object's velocity over some period of time, you can calculate its average angular acceleration. We are asked to find the number of revolutions. What is its angular acceleration ? The tangent is a line that is perpendicular to the radius at that point. Final angular velocity = zero. To learn more, including how to calculate average angular acceleration, read on. alpha = (89.3rad/s - 0rad/s) / 40s = 2.23rad/s^2 (there's your answer for angular acceleration) To find the number of revolutions made by the wheels, you need to know the total angle rotated by the wheel during the take-off (e.g. After making 2870 revolutions, its angular speed is 137 rad/s. A particle starts from it's initial velocity of 4pay rad/min and its final velocity of 24pay rad/min and its cover's displecemet of 3pay rad. When the object is moving in a circle, such as a spinning tire or a rotating CD, velocity and acceleration are generally measured by the angle of rotation. If you really can’t stand to see another ad again, then please consider supporting our work with a contribution to wikiHow. 500 rev; B. Negative motion is measured in a clockwise direction. Solution for The angular speed of a disk decreases uniformly from 12 to 5 rad/s in 16 s . Where; ω = Angular velocity N = Number of revolutions per minute. Average Speed/Velocity=2πr/T where, T is the period of the system and r is the radius of the revolution. (a) The wheel initially rotates 1 revolution every 8 seconds, or 0.125 rev/s. In two dimensions, angular velocity is a number with plus or minus sign indicating orientation, but not pointing in a direction. Thus, the angular acceleration for this spinning object when it has been spinning for 6.5 seconds is 78.0 radians per second squared. There are 12 references cited in this article, which can be found at the bottom of the page. Calculate the angular acceleration. Your support helps wikiHow to create more in-depth illustrated articles and videos and to share our trusted brand of instructional content with millions of people all over the world. Question: A Find The Angular Acceleration In Revolutions Per Second Per Second.B Find The Number Of Revolutions Made By The Fan Blades During The Time That They Are Slowing Down In Part AC How Many Mkre Seconds Are Required For The Fan To Come To Rest If The Angular Acceleration Remains Constant At The Value Calculated In Part A? Consider a compact disc at the moment you place it in the CD player. = 637 rad/2π = 100 revs. Find its angular acceleration. Find the initial angular velocity in units of rad/s and the number of revolutions the wheel rotates through before it comes to rest. In mathematical and physical terms, however, only the word “acceleration” is used. ω … The flywheel is given a suitable number (n) of rotation so that the cord is wound round the axle without overlapping. Remember to express final results with the proper units. I found the angular acceleration to be 20.1 rad/s^2. The speed of a wheel increases from 600 rpm to 1200 rpm in 20 s. Be careful of the sign of the angular acceleration. A turbine started from rest to 180 rpm in 6 minutes at a constant acceleration. Angular acceleration = -3 radians per second2 Initial angular velocity = 6 radians per second. ω=2π/T=2πf where, f is frequency and T is the period Unlike tangential velocity, angular velocity of all points on the platform doing circular motion are equal to each other since the number of rotations per unit time are equal. Ð°Ð²Ð¸ÑÐ¸Ð¼ÑÐ¼ Ð² ÐÐ¸ÐµÐ²Ðµ, Draw the corresponding position-versus-time graph, ÐÐµÑÐµÐ²Ð¾Ð´Ð¸ÑÐµ ÑÐ¾ÑÑÑÐ´Ð½Ð¸ÐºÐ¾Ð² Ð½Ð° ÑÐ´Ð°Ð»ÐµÐ½ÐºÑ, Ð½Ð¾. To solve for v, from above, v = ωr, where ω = π/2 and r = 10m: (π/2) (10) = 5π rad/s = 15.7 m/s The angular speed of a disk decreases uniformly from 12 rad/s to 4 rad/s in 16 seconds. Find the number of revolutions that it makes within the elapsed time. Multiply that by 0.195 sec to get the number of revs. ANSWER (b) First calculate the rotation angle. http://physics.tutorvista.com/motion/angular-acceleration.html, http://www.softschools.com/formulas/physics/angular_acceleration_formula/153/, consider supporting our work with a contribution to wikiHow, For the example illustrated above, studies have led to the function, In this example, find the first derivative of the position function, If desired, this function could be used to calculate the angular velocity of the spinning object at any desired time, In the previous step, you used the function for position to find the angular velocity, For the sample problem in the illustration, suppose you know that the function for the position of the spinning object is. Compute (a) the angular acceleration and (b) the number of revolutions made in this time? w2 = final angular velocity = 0 (when rotor stops) w1 = 1.08 x 10^4 * 2(pi) = 3.18(pi) x 10^4 rad/sec. This article has been viewed 66,198 times. If you recall that a full circle is 360 degrees, you can find the conversion as follows: Thus, one radian is about equal to 57.3 degrees. Average acceleration Recall that if the angular velocity decreases uniformly, the angular acceleration will remain constant. How can you find angular acceleration in revolutions per second squared? What are the formulas to find the initial acceleration of an object? Every day at wikiHow, we work hard to give you access to instructions and information that will help you live a better life, whether it's keeping you safer, healthier, or improving your well-being. Determine the function for angular position. What is the direction of radial and tangential acceleration and how do they affect each other? Angular velocity is expressed in radians per time. (b) How many more seconds are required for the fan to come People often use the word “acceleration” to mean speeding up, and “deceleration” to mean slowing down. Therefore, a half circle is π radians, and a quarter circle is π/2 radians. They are then called angular velocity and angular acceleration. Tangential acceleration means the straight line direction of the tangent at some measured point along the circle. How To Calculate The Angular Velocity Formula. The formula for calculating angular velocity: ω = 2πN / 60. The angle that is being measured is commonly represented by. Part A Find the angular acceleration in revolutions per second per second. We know ads can be annoying, but they’re what allow us to make all of wikiHow available for free. This will be its final angular velocity. International System of Units. But I dont know how to find the number of complete revolutions needed to stop. An electric fan is turned off, and its angular velocity decreases uniformly from 500.0 rev/min to 200.0 rev/min in 4.00 s. (a) Find the angular acceleration in rev/s' and the number of revolutions made by the motor in the 4.00 s interval. _____ rev Facebook. To learn more, including how to calculate average angular acceleration, read on. Now that you have found the angular acceleration, use the formula (w2)^2 - (w1)^2 = 2(α)(Θ) where. If you know the acceleration in radians per second squared, divide that answer by 6.28 to get revolutions per second squared. Here, we are asked to find the number of revolutions. Your speed is the angular velocity multiplied by your distance from the center of the wheel: v = r ω = 4 * 0.785 = 3.14 m/s Therefore, the angular acceleration is just the total change in angular velocity divided by the total change in time. In that sense is related to frequency but in terms of how many times it turns a full period of motion in radians units. Most people have a general understanding of the idea of velocity and acceleration. With this information, you can calculate its angular acceleration at any chosen instant. ~~~~~ (a) the angular acceleration = = rad/s^2 = rad/s^2 = rad/s^2 = 0.5 rad/s^2. % of people told us that this article helped them. How many revolutions will it make during this time ? You have correctly got the final angular velocity to be 0.474 rev/s. Angular (or radial) measurements are generally counterclockwise. Initial acceleration generally has to be given as a condition of the problem or the experiment. What is its angular acceleration ? The owner's manual for the CD player provides the information that the CD reaches its playing speed in 4.0 seconds. The number of revolutions that the shaft will make during this time is 4:19 13.9k LIKES. α = angular acceleration that you have determined (this must have a negative sign since it is actually slowing down) Θ = number of revolutions Let’s solve an example; Find the Angular Velocity with a number of revolutions per minute as 60. Calculating Instantaneous Angular Acceleration, {"smallUrl":"https:\/\/www.wikihow.com\/images\/thumb\/d\/da\/Calculate-Angular-Acceleration-Step-1-Version-2.jpg\/v4-460px-Calculate-Angular-Acceleration-Step-1-Version-2.jpg","bigUrl":"\/images\/thumb\/d\/da\/Calculate-Angular-Acceleration-Step-1-Version-2.jpg\/aid4281265-v4-728px-Calculate-Angular-Acceleration-Step-1-Version-2.jpg","smallWidth":460,"smallHeight":345,"bigWidth":"728","bigHeight":"546","licensing":"


text:
# My Bad Code (Or: How to check if you are a power of two properly)

I was looking at some past programming I had done, and I had a problem where I wanted to detect if a given number was a power of two.

I did this by doing the following test:

$\left|\left[\frac{\log(x)}{\log(2)}\right]-\frac{\log(x)}{\log(2)}\right|<\epsilon$

That is, I took the log of x to the base two, rounded it, subtracted it from the non-round version, took the absolute value and checked if it was small.

To be fair, it works. A value of epsilon can be found so that it will work on even the worst floating point unit as it is known that x is an integer. However, it still requires a subtraction, an absolute value, a division and a calculation of log.

This is all particularly stupid: You shouldn’t use floating point numbers to show something about an integer. That is just crazy.

Essentially what we need to check is if the binary expansion of x has just one on-bit.

Remember what happens to the binary expansion of a (non-zero) number when you subtract one from it. It flips all the zeroes on the right into ones until the first one, which it turns into a zero. For instance, 0101000 becomes 0100111. Now this is useful, because if we bitwise ‘and’ this number with the original we will just get the number with the first one taken away (as each bit from the start to that position has changed in x-1). For instance, from 0101000 we would get 0100000.

The important thing to note then is that if it started off as a power of two, then it it will equal 0 in the end. That is, we can test if a (non-zero) number is a power of two by checking if:

$x \:\&\: (x-1) = 0$

This test also accepts zero, but this worked brilliantly for me: Zero was a valid input too.

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