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Review question

# Can we show these two circles lie entirely outside each other? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R6390

## Question

Prove that the points whose coordinates satisfy the equation $x^2+y^2+2gx+2fy+c=0$ lie on a circle. State the coordinates of the centre of the circle and the length of its radius.

Prove that the circles $x^2+y^2-20x-16y+128=0$ and $4x^2+4y^2+16x-24y-29=0$ lie entirely outside each other, and find the length of the shortest distance from a point on one circle to a point on the other.


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The interpretation of oligonucleotide array experiments depends on the quality of the target cRNA used. cRNA target quality is assessed by quantitative analysis of the representation of 5' and 3' sequences of control genes using commercially available Test arrays. The Test array provides an economically priced means of determining the quality of labeled target prior to analysis on whole genome expression arrays. This manuscript validates the use of a duplex RT-PCR assay as a faster (6 h) and less expensive (<US$10) but equally accurate alternative to the Test arrays in determining biotinylated cRNA quality. Forty-one different cRNA samples were hybridized to HG-U133A microarrays from Affymetrix. Ten cRNA samples with a ß-actin 3'/5' ratio >6 were chosen and classified as degraded cRNAs, and 31 samples with a ß-actin 3'/5' ratio <6 were selected as good quality cRNAs. Blinded samples were then used for the RT-PCR assay. After gel electrophoresis, optical densities of the amplified 3' and 5' fragments of ß-actin were measured and the 3'/5' ratio was calculated. There was a strong correlation (r² = 0.6802) between the array and the RT-PCR ß-actin 3'/5' ratios. Moreover, the RT-PCR 3'/5' ratio was significantly different (P < 0.0001) between undegraded (mean ± SD, 0.34 ± 0.09) and degraded (1.71 ± 0.83) samples. None of the other parameters analyzed, such as i) the starting amount of RNA, ii) RNA quality assessed using the Bioanalyzer Chip technology, or iii) the concentration and OD260/OD280 ratio of the purified biotinylated cRNA, correlated with cRNA quality. Microarray; Biotinylated cRNA; Probe quality; 5' Representation; 3'/5' Ratio Braz J Med Biol Res, May 2006, Volume 39(5) 589-593 (Short Communication) Rapid, reliable and inexpensive quality assessment of biotinylated cRNA T. Zander1,2, J.A. Yunes1,3, A.A. Cardoso1 and L.M. Nadler1 1Department of Medical Oncology, Dana-Farber Cancer Institute, Harvard Medical School, Boston, MA, USA 2Molecular Tumor Biology and Tumor Immunology, Center for Internal Medicine, University Cologne, Cologne, Germany 3Centro Infantil Boldrini, Campinas, SP, Brasil Text References Correspondence and Footnotes # Abstract The interpretation of oligonucleotide array experiments depends on the quality of the target cRNA used. cRNA target quality is assessed by quantitative analysis of the representation of 5' and 3' sequences of control genes using commercially available Test arrays. The Test array provides an economically priced means of determining the quality of labeled target prior to analysis on whole genome expression arrays. This manuscript validates the use of a duplex RT-PCR assay as a faster (6 h) and less expensive (<US$10) but equally accurate alternative to the Test arrays in determining biotinylated cRNA quality. Forty-one different cRNA samples were hybridized to HG-U133A microarrays from Affymetrix. Ten cRNA samples with a ß-actin 3'/5' ratio >6 were chosen and classified as degraded cRNAs, and 31 samples with a ß-actin 3'/5' ratio <6 were selected as good quality cRNAs. Blinded samples were then used for the RT-PCR assay. After gel electrophoresis, optical densities of the amplified 3' and 5' fragments of ß-actin were measured and the 3'/5' ratio was calculated. There was a strong correlation (r2 = 0.6802) between the array and the RT-PCR ß-actin 3'/5' ratios. Moreover, the RT-PCR 3'/5' ratio was significantly different (P < 0.0001) between undegraded (mean ± SD, 0.34 ± 0.09) and degraded (1.71 ± 0.83) samples. None of the other parameters analyzed, such as i) the starting amount of RNA, ii) RNA quality assessed using the Bioanalyzer Chip technology, or iii) the concentration and OD260/OD280 ratio of the purified biotinylated cRNA, correlated with cRNA quality.

Key words: Microarray, Biotinylated cRNA, Probe quality, 5' Representation, 3'/5' Ratio

Analysis of the transcriptome using high-density oligonucleotide arrays is an important tool for discovery in biology (1). The preparation of the target cRNA is a multi-step procedure that requires RNA isolation, first strand cDNA synthesis with an oligo(dT) primer bearing a T7 promoter, second-strand cDNA synthesis, and a final in vitro transcription using T7 RNA polymerase and biotinylated ribonucleotides. These enzymatic reactions are sensitive to the quality of the starting RNA, the intermediate cDNA template, the occurrence of endogenous priming, and the presence of impurities, such as RNases. Incomplete cDNA or cRNA synthesis usually results in 5' shortening of the target cRNA molecules (2), which is difficult to assess using gel-based assays. To overcome this obstacle, microarray's oligonucleotide probes are usually selected among the last 600 nucleotides of the mRNA end, and cRNA target quality is assessed by quantitative analysis of the representation of 5' and 3' sequences of control genes, such as ß-actin, GAPDH, ISGF3, RGE, or TFRR. For quality control, Test arrays containing probe sets corresponding to the 5' and 3' regions of selected genes are generally used (e.g., TEST3 Chip; Affymetrix, Santa Clara, CA, USA). Here, the main quality criterion is the 3'/5' ratio. However, for 3'/5' ratio analysis, array technology may not be the most effective method and can be replaced by other quantitative approaches. RT-PCR has been successfully applied for semi-quantification (3) and quantification of RNA templates (4), and in a multiplex setting (5). Its comparability to microarray data has been demonstrated (6,7). Since no dramatic changes in expression values from sample to sample were observed (8) and degradation of different RNA species is similar and correlates with the overall RNA degradation (9), analysis of ubiquitous housekeeping genes is a valid surrogate for overall degradation and incomplete cDNA/cRNA synthesis. RT-PCR permitted the confirmation of the loss of 5' complexity in RNA subjected to autolysis (10). However, to our knowledge, no similar quality assessment of biotinylated cRNA by RT-PCR has been described. Here, we report a rapid, reliable and simple RT-PCR-based method for quality assessment of biotinylated cRNA.

cRNA was generated from total RNA from human blood mononuclear cells following standard protocols (11). The Institutional Review Board approved these studies and informed consent was obtained from donors. Hybridization, washing, staining, and scanning of HG-U133A Affymetrix arrays were performed according to the Affymetrix instructions. Target cRNA quality was assessed by calculating the 3'/5' ratio value for the ß-actin gene (interrogation probes: AFFX-HSAC07/X00351_5_at and AFFX-HSAC07/X00351_3_at). A 3'/5' ratio of 6 was defined as the cut-off value to discriminate the poor quality target cRNAs. From our existing database, 10 samples with a 3'/5' ratio >6 were chosen and classified as degraded cRNAs, and 31 samples with a 3'/5' ratio of ß-actin <6 were selected as good quality cRNAs. Blinded samples were then used for the RT-PCR assay and the code was opened only after results were obtained and interpreted.

Primers were designed using the OLIGO 6.7 software (Molecular Biology Insights, Cascade, CO, USA): 204U, 5'-TGGTGGGC ATGGGTCAGA-3', and 453L, 5'-GTACA TGGCTGGGGTGTTGA-3' for the 5' ß-actin region (269 bp), and 1260U, 5'-AAC AAGATGAGATTGGCA-3', and 1696L, 5'-GACCAAAAGCCTTCATACAT-3' for the 3' region (456 bp), as shown in Figure 1B. For the duplex RT-PCR analysis, 500 ng of the biotinylated target cRNA (not fragmented) plus 1 µg of random hexamers were denatured for 2 min at 70ºC in a total volume of 12.5 µL, and then chilled on ice. Reverse transcription reactions were carried out in 20 µL containing 50 mM Tris-HCl, pH 8.3, 75 mM KCl, 3.5 mM MgCl2, 0.5 mM dNTPs, 20 U recombinant RNase inhibitor (BD Bioscience Clontech, Palo Alto, CA, USA) and 200 U MMLV reverse transcriptase (BD Bioscience Clontech) at 42ºC for 1 h followed by 5-min heating at 80ºC to denature proteins. The cDNA thus obtained was diluted to a total volume of 100 µL with H2O and 3 µL was used for PCR. PCR was performed in 30 µL containing 50 mM KCl, 10 mM Tris-HCl, pH 8.3, 3.5 mM MgCl2, 0.1% Triton X100, 0.2 mM dNTPs, 1 U Taq polymerase (Fisher Scientific, Pittsburgh, PA, USA), 5 pmol of each primer, in a GeneAmp 9600 thermocycler (Applied Biosystems, Foster City, CA, USA). The cycling protocol consisted of 10 min at 95ºC for denaturation, followed by 6 cycles of annealing at decreasing temperatures from 65º to 60ºC (1ºC decrease every cycle) for 45 s, extension at 72ºC for 45 s and denaturation at 95ºC for 30 s. Subsequently, 19 cycles were performed with annealing at 59ºC for 45 s, extension at 72ºC for 45 s, and denaturation at 95ºC for 30 s. A final extension step for 5 min at 72ºC was included. Eight microliters of the PCR products was analyzed on a 2% agarose gel and visualized with ethidium bromide. For further semiquantitative analysis, densitometry was performed using the Kodak 1-D Image Analysis Software (Kodak, New Haven, CT, USA). The Student t-test and the Pearson correlation coefficient were applied to assess the significance of the results.

As described above, a duplex RT-PCR analysis was employed to amplify two PCR products corresponding to the 3' and 5' regions of ß-actin, predicted as 456 and 269 bp, respectively. The expected PCR products were detected in 40 of 41 target cRNA samples tested (Figure 1A). Sample 23 showed very weak amplification products even after several attempts. Since PCR conditions favor the amplification of the 5' ß-actin region, higher amounts of the 269-bp product were expected in cRNAs of the highest quality (e.g., samples 2, 3, and 26). A similar pattern was also observed for high-quality total RNA (data not shown). By visual inspection, nine samples were clearly distinguished as having a higher amount of the 3' region (456 bp) than the 5' region (269 bp) PCR product. Measurement of all bands by optical density and calculation of their 3'/5' ratios, resulted in their clustering into two distinct groups. Most importantly, there was a strong correlation (r2 = 0.6802) between the array (CHIP) and the RT-PCR ß-actin 3'/5' ratios (Figure 1C). As expected, all samples classified as degraded cRNAs by the array data were clustered into the RT-PCR higher ratio group. Furthermore, the 3'/5' ratio calculated by RT-PCR was significantly different (P < 0.0001) between the undegraded (mean ± SD, 0.34 ± 0.09) and degraded (1.71 ± 0.83) group as defined by the array data (Figure 1D).

With the exception of one sample (sample 20), all other degraded samples had a 3'/5' RT-PCR ratio above 0.95. Sample 20 had a 3'/5' RT-PCR ratio of 0.78. All undegraded samples presented a 3'/5' RT-PCR ratio lower than 0.47. Therefore, an empirical cut-off point for the RT-PCR assay could be set at 0.65.

As shown in Table 1, the quality of the cRNA obtained was not associated with the starting amount or quality of RNA used, as nearly all samples (except one; sample 23) were qualified as excellent using the Bioanalyzer Chip technology (Agilent, Palo Alto, CA, USA). Likewise, the concentration and OD260/OD280 ratio of the purified biotinylated cRNA are not related to the quality of the cRNA. Finally, we could not assess the cRNA quality using gel-based systems since evaluation of smear shapes is very subjective (data not shown).

In summary, this fast, inexpensive and simple RT-PCR predicted the quality of the cRNA comparable to the 3'/5' ratio of the Affymetrix test arrays. None of the other parameters analyzed correlated with the 3'/5' ratio of the cRNA.

Figure 1.
cRNA quality assessment by the analysis of ß-actin 5' and 3' sequence abundance using a duplex RT-PCR in comparison to the microarray-obtained 3'/5' ratio. A, RT-PCR products (sample 1 through 41) were analyzed on a 2% agarose gel stained with ethidium bromide. The 3' ß-actin (456 bp) and the 5' ß-actin (269 bp) PCR products were detected. L = 1 Kb DNA ladder (Invitrogen, Carlsbad, CA, USA). Arrows show samples classified as degraded by the microarray result. B, ß-actin cDNA sequence. ATG and TGA correspond to the translation start and termination codons, respectively. Sequence regions targeted by the array 3' and 5' probes are depicted. Underlined sequences correspond to the location of primers used for PCR. C, The optical density of each band was measured and the 3'/5' ratio was calculated and plotted against the respective 3'/5' ratio assessed by the Affymetrix HG-U133A microarray (CHIP) analysis. CHIP 3'/5' ratios were transformed into logarithmic values for correlation analysis (Pearson test). D, The RT-PCR 3'/5'ratios of samples classified as undegraded (CHIP 3'/5' ratio <6) and degraded (CHIP 3'/5' ratio >6) are shown. Significance was calculated using the Student t-test.

Table 1.
Summary of the parameters tested to characterize biotinylated cRNA quality.

# Acknowledgments

We thank Dr. Edward Alwin Fox (Dana-Farber Cancer Institute, Boston, MA, USA) for numerous helpful discussions and for providing the cRNA samples.

Address for correspondence: J.A. Yunes, Centro Infantil Boldrini, Rua Gabriel Porto, 1270, 13083-210 Campinas, SP, Brasil. Fax: +55-19-3289-3571. E-mail: andres@boldrini.org.br

T. Zander was supported in part by the Frauke Weiskam-Christel Ruranski Foundation (Germany), and by grants from the NIH (PO1-CA78378), and the Leukemia and Lymphoma Society (6447) to Joachim Schultze. J.A. Yunes and A.A. Cardoso were partially supported by a grant from NIH (PO1-CA68484) and J.A. Yunes was partially supported by a grant from the Children's Leukemia Research Association (USA). Publication supported by FAPESP. Received July 29, 2005. Accepted January 2, 2006.

• 1. Pomeroy SL, Tamayo P, Gaasenbeek M et al. (2002). Prediction of central nervous system embryonal tumour outcome based on gene expression. Nature, 415: 436-442.
• 2. Swift GH, Peyton MJ & MacDonald RJ (2000). Assessment of RNA quality by semi-quantitative RT-PCR of multiple regions of a long ubiquitous mRNA. BioTechniques, 28: 524, 526, 528, 530-531.
• 3. Feldhahn N, Schwering I, Lee S et al. (2002). Silencing of B cell receptor signals in human naive B cells. Journal of Experimental Medicine, 196: 1291-1305.
• 4. Roth CM (2002). Quantifying gene expression. Current Issues in Molecular Biology, 4: 93-100.
• 5. Burgess SC & Davison TF (1999). A quantitative duplex PCR technique for measuring amounts of cell-associated Marek's disease virus: differences in two populations of lymphoma cells. Journal of Virological Methods, 82: 27-37.
• 6. Gnatenko DV, Dunn JJ, McCorkle SR et al. (2003). Transcript profiling of human platelets using microarray and serial analysis of gene expression. Blood, 101: 2285-2293.
• 7. Islam TC, Lindvall J, Wennborg A et al. (2002). Expression profiling in transformed human B cells: influence of Btk mutations and comparison to B cell lymphomas using filter and oligonucleotide arrays. European Journal of Immunology, 32: 982-993.
• 8. Warrington JA, Nair A, Mahadevappa M et al. (2000). Comparison of human adult and fetal expression and identification of 535 housekeeping/maintenance genes. Physiological Genomics, 2: 143-147.
• 9. Tong D, Schneeberger C, Leodolter S et al. (1997). Quantitative determination of gene expression by competitive reverse transcription-polymerase chain reaction in degraded RNA samples. Analytical Biochemistry, 251: 173-177.
• 10. Sugita M, Haney JL, Gemmill RM et al. (2001). One-step duplex reverse transcription-polymerase chain reaction for quantitative assessment of RNA degradation. Analytical Biochemistry, 295: 113-116.
• 11. Golub TR, Slonim DK, Tamayo P et al. (1999). Molecular classification of cancer: class discovery and class prediction by gene expression monitoring. Science, 286: 531-537.
• Correspondence and Footnotes

# Publication Dates

• Publication in this collection
20 Apr 2006
• Date of issue
May 2006

• Accepted
02 Jan 2006

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# How does experienced pressure vary with airspeed?

Drag increases with the square of airspeed. Is the same true of the pressure that a forward-facing area experiences?

As an aircraft approaches and then exceeds the speed of sound, how does experienced pressure on a forward-facing area increase?

• – Ryan Mortensen Jun 2 '16 at 5:44
• I don't mean to nit pick, but this is really should be two separate questions. – Ryan Mortensen Jun 2 '16 at 5:49

1) Pressure is simply a force divided by an area, so yes: The dynamic pressure experienced by a forward-facing surface will, in an incompressible flow, vary with the square of the speed.

2) The dynamic pressure increases dramatically as one approaches the speed of sound, once going faster than mach one the pressure will begin to decrease. The trans-sonic phase (around mach one) is difficult to describe and can depend on the airflow around the body - but in many cases it can be quite stable.

Drag increases with the square of airspeed. Is the same true of the pressure that a forward-facing area experiences?

Drag increases with the square of the airspeed because it is a function of the dynamic pressure $\bar{p}$:

$$D = C_D A \bar{p} = C_D A \frac{1}{2} \rho V^2$$

So the answer is yes, because the drag is a function of the (dynamic) pressure that a forward-facing area experiences.

I do not speak of the static pressure because that is a function of the atmospheric conditions, and is not a function of the movement of the aircraft through the atmosphere.

As an aircraft approaches and then exceeds the speed of sound, how does experienced pressure on a forward-facing area increase?


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# You're reading: Posts Tagged: Michael Atiyah

### Aperiodical News Roundup – June 2021

Here’s a round-up of mathematical things that happened in June, and things you might want to know about that are happening in the future!

## News

#### News In Brief

• YouTuber and PhD physicist Derek Muller (Veritasium) has recently been involved in a physics-off with UCLA professor Alexander Kusenko, when they disagreed over the explanation behind a physical phenomenon, which escalated to a $10,000 bet over who was right. Long story short, Veritasium won the bet (as covered in this IFLScience news story) and will be using the money to fund a science communication contest. If you’ve got an under-a-minute maths/science video you can post on YouTube or TikTok, you could win a prize of up to$5,000. Props to Derek for encouraging more STEM communication and promoting new talent!
• It’s been formally announced that Neil Sloane is stepping down as president of the OEIS – Russ Cox will take over presidential duties, while Sloane steps down to Chairman of the Board so he can dedicate more time to his writing projects (which we’re assured ‘naturally involve sequences’). Cox has been involved with the OEIS for over 25 years and has been a major contributor to the backend software that makes the site run, so he’s a safe pair of hands to take the project on.
• The eleven 2021 LMS Prize winners were announced at the Society’s Meeting on 2nd July, and the prizes recognise contributions to mathematics in a variety of areas. (via @LondMathSoc)

#### Alan Turing £50 note launches

On 23rd June the new Alan Turing £50 note was launched, featuring an image of Turing, a quote and various mathematical diagrams. Bletchley Park marked the occasion with a #Turing50Takeover, and the Bank of England has a whole page of info about the new polymer note on their website.

Meanwhile, in Turing-adjacent news, the National Museum of Computing has launched an online Virtual Enigma machine you can use to simulate the device behind the famous Enigma code, along with a video explaining the machine. This joins a host of other virtual historical computers they’ve built, including the Colossus that cracked the code, the Lorenz machine and even ERNIE the random number generator!

#### Claimed proof of Riemann Hypothesis

Another claimed proof of the Riemann Hypothesis, this time by Kumar Easwaran, emerged this month, and since like all big claims it would need thorough checking before acceptance by the mathematical community, there was some initial skepticism. (This didn’t stop the media from latching on to it as an exciting story though). Since claimed proofs of Riemann are like buses, many mathematicians don’t give them much attention, but Alex Kontorovich took the time to thoroughly debunk this one to save you the trouble.

If you want some actual Riemann Hypothesis news, here’s some: John Baez reports that Alain Connes and Caterina Consani have made some potential progress on part of the problem. In the words of Baez, “my interest is piqued”.

Thuses is a website for mathematicians to publish and discuss ideas of interest to the mathematical community. It’s described as “a perfect place to share new approaches, slick proofs, and surprising counterexamples. A place for ‘folklore results’ that are considered known but don’t actually exist in literature. A place for everything in math that just has to be shared.” (via Piper H)

The Royal Society has published a set of papers on modelling that shaped the early COVID-19 pandemic response in the UK as a special journal issue that’s free to access.

The BSHM (British Society for the History of Mathematics) has launched the Bibby Awards in the History of Mathematics, for “contributions to the popularization of the history of mathematics in education”. Named after (and funded by the legacy of) the late BSHM member Neil Bibby, up to four awards of £400 can be made each academic year, in return for which holders are expected to give two free talks in schools and produce four digital resources (videos, PDFs or interactives) for the BSHM website. (via Sarah Hart)

## Events

The Isaac Newton Institute in Cambridge is hosting The unity of mathematics: A conference in honour of Sir Michael Atiyah which will take place in September 2021 as a hybrid event with a mixture of in-person and virtual talks. The closing date for registration for physical participants is 8th August.

There’s just about still time to register for the People, Places, Practices History of Maths Conference (registration closes 9th July) taking place 12-15 July online (coordinated by the University of St. Andrews). With around 90 speakers contributing, the programme looks packed, and talks will be available to watch ahead, or at the specified time to be followed by a live Q&A.

Alexandre Borovik reports on his blog about Azat Miftakhov day, an event organised online by the Azat Miftakhov committee in solidarity with Azat Miftakhov – a graduate student from Moscow State University who was sentenced to six years in a medium-security penal colony and has already been arbitrarily detained by Russian state authorities for almost two and a half years. Fields medalist Cedric Villani made a speech at the event, and you can watch videos from the event on YouTube.

### Sir Michael Atiyah has died

Over the weekend we heard the sad news that mathematician Sir Michael Atiyah has passed away on 11th January. One of the few mathematicians to have been awarded both a Fields medal and an Abel prize, Atiyah leaves behind an extensive mathematical legacy and will be missed by many.

A tribute to former President of the Royal Society Sir Michael Atiyah OM FRS (1929 – 2019), on the Royal Society website

Mathematician Sir Michael Atiyah dies aged 89, at BBC News

Michael Atiyah, Mathematician in Newton’s Footsteps, Dies at 89 at the New York Times

### Atiyah-Riemann Proof: banter summary

Today the internet has been getting excited about Sir Michael Atiyah’s claimed proof of the Riemann Hypothesis, which he presented at the Heidelberg Laureate Forum this morning. We’ve collected all the relevant links and tweets to help you make sense of what’s going down in critical-line-town.

### HLF Blogs: What is the Riemann Hypothesis?

This week, Katie and Paul are blogging from the Heidelberg Laureate Forum – a week-long maths conference where current young researchers in maths and computer science can meet and hear talks by top-level prize-winning researchers. For more information about the HLF, visit the Heidelberg Laureate Forum website.

This year at the HLF there are multiple sessions in the program concerning the Riemann Hypothesis, including a talk from one of the laureates, and one of the young-researcher-led workshop sessions. But what exactly is the Riemann Hypothesis, and what is its place in mathematics?

### Michael Atiyah claims proof of Riemann Hypothesis

The internet is buzzing with the news: algebraic topologist and geometer Sir Michael Atiyah has announced that at his talk next Monday at the Heidelberg Laureate Forum, he will deliver a simple new proof of the long-standing Riemann Hypothesis. The news comes as a surprise to many, considering that since the result has been unproven since its conjecture in 1859, if a proof does exist it will likely be far from simple.

### Blogs from this year’s Heidelberg Laureate Forum

Heidelberg Castle selfie

Paul and I have spent this week blogging from the Heidelberg Laureate Forum, an international event for PhD/postdoc students and top-level maths and computer science researchers.

It was a long week of extravagant dinners, incredible talks and press conferences, (maths) celeb spotting, branded conference freebies, hilarious quotes and exceptional hospitality. Oh, and blogging. Here’s a round-up of what we wrote, in case you’ve missed it this week, as well as some of the other posts the rest of the HLF blog team wrote.

### Heidelberg Laureate Forum 2017

This week our roving reporters Katie and Paul have gone on a trip to Heidelberg in Germany, where the world’s foremost undergraduate, masters, PhD and postdoc students in maths and computer science are gathering for the fifth annual Heidelberg Laureate Forum.


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## Sunday, 28 July 2013

### An alternative variational principle

In the previous post I talked about one of the less quoted results in the variational theory of Sturm-Liouville eigenvalue problems which I owe to the book on differential equations by Erich Kamke and which I reworked for the purposes of my dissertation.

The other result that I also found in Kamke's book states an alternative variational principle which can be used to obtain estimates of the eigenvalues. This method gives a weaker upper bound than the Rayleigh quotient, however it requires smaller computational effort. In return, one can widen the class of trial functions without severely complicating the calculations.

$$J\left\{ \phi\right\}=\frac{\left\langle \phi,L\left[\phi\right]\right\rangle }{\left\langle \phi,\phi\right\rangle }$$
Then using Cauchy-Schwarz inequality we obtain.
$$\left(\left\langle \phi,L\left[\phi\right]\right\rangle \right)^{2}\leq\left\langle \phi,\phi\right\rangle \left\langle L\left[\phi\right],L\left[\phi\right]\right\rangle \frac{\left\langle \phi,L\left[\phi\right]\right\rangle }{\left\langle \phi,\phi\right\rangle }\leq\frac{\left\langle L\left[\phi\right],L\left[\phi\right]\right\rangle }{\left\langle \phi,L\left[\phi\right]\right\rangle }$$
Written explicitly
$$\frac{\int_{\Omega}\rho\phi L\left[\phi\right]d\boldsymbol{x}}{\int_{\Omega}\rho\phi^{2}d\boldsymbol{x}}\le\frac{\int_{\Omega}\rho\left(L\left[\phi\right]\right)^{2}d\boldsymbol{x}}{\int_{\Omega}\rho\phi L\left[\phi\right]d\boldsymbol{x}}$$
Hence we can replace the original problem with the following one
$$K\left\{ \phi\right\} =\frac{\left\langle L\left[\phi\right],L\left[\phi\right]\right\rangle }{\left\langle \phi,L\left[\phi\right]\right\rangle }\to\min,\qquad\phi\in\mathfrak{A}$$
Example.
Consider the eigenvalue problem
$$-y''=\lambda y,\quad y\left(0\right)=y\left(1\right)=0$$
Take the trial function
$$y_{1}=\frac{x}{12}-\frac{x^{3}}{6}+\frac{x^{4}}{12}$$
First we perform the calculation using the Rayleigh quotient
$$J\left\{ y_{1}\right\} =\frac{\left\langle y_{1},L\left[y_{1}\right]\right\rangle }{\left\langle y_{1},y_{1}\right\rangle }=-\frac{\int_{0}^{1}y_{1}y_{1}''dx}{\int_{0}^{1}y_{1}^{2}dx}$$
\begin{aligned}\int_{0}^{1}y_{1}y_{1}''dx & =\int_{0}^{1}\left[\left(\frac{x}{12}-\frac{x^{3}}{6}+\frac{x^{4}}{12}\right)\left(-x+x^{2}\right)\right]dx\\ & =\int_{0}^{1}\left(\frac{x^{6}}{12}-\frac{x^{5}}{4}+\frac{x^{4}}{6}+\frac{x^{3}}{12}-\frac{x^{2}}{12}\right)dx\\ & =-\frac{17}{5040} \end{aligned}
The factor evaluated above will be common to both variational quotients
\begin{aligned}\int_{0}^{1}y_{1}^{2}dx & =\int_{0}^{1}\left(\frac{x^{8}}{144}-\frac{x^{7}}{36}+\frac{x^{6}}{36}+\frac{x^{5}}{72}+\frac{x^{2}}{144}\right)dx\\ & =\frac{31}{90720} \end{aligned}
$$\lambda_{1}\le J\left\{ y_{1}\right\} =\frac{17\cdot90720}{5040\cdot31}\approx9.871$$
Now we use the alternative variational principle
$$K\left\{ y_{1}\right\} =\frac{\left\langle L\left[y_{1}\right],L\left[y_{1}\right]\right\rangle }{\left\langle y_{1},L\left[y_{1}\right]\right\rangle }=-\frac{\int_{0}^{1}y_{1}''^{2}dx}{\int_{0}^{1}y_{1}y_{1}''dx}$$
Obviously, the announced gain in computational efficiency comes from replacing $$y^{2}$$ with $$\left(L\left[y\right]\right)^{2}$$ which will be a polynomial of the order less by 4, than $$y^{2}$$.
$$\int_{0}^{1}y_{1}''^{2}dx=\int_{0}^{1}\left(x^{4}-2x^{3}+x^{2}\right)dx=\frac{1}{30}$$
$$\lambda_{1}\le K\left\{ y_{1}\right\} =\frac{5040}{17\cdot30}\approx9.882$$
Hence
$$K\left\{ y_{1}\right\} >J\left\{ y_{1}\right\} >\lambda_{1}=\pi^{2}$$
as expected, however $$K\left\{ y_{1}\right\}$$ takes less operations to evaluate.


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# Angle between vector and tangent vector

Tags:
1. Jan 24, 2017

### Prof. 27

1. The problem statement, all variables and given/known data
My problem is:

For the logarithmic spiral R(t) = (e^t cost, e^t sint), show that the angle between R(t) and the tangent vector at R(t) is independent of t.

2. Relevant equations
N/A

3. The attempt at a solution
The tangent vector is just the vector that you get when you take the derivative of each element of the vector so:

R(t) = (e^t cost, e^t sint)
R'(t) = (e^t*cost-e^t*sint, e^t*sint + e^t*cost)

First I tried to show that the dot product was zero using the multiply the x's and y's then add method. This would imply that the angle between the vector and tangent vector was always ninety degrees; unfortunately as can be seen from an inspection of the two vectors, the dot product is e^2t.

Then I went the long route and used the dot product formula (a dot b = ||a||*||b||*cos(@)) to try to calculate a single angle @ that held constant between them. This was also a bust.

e^2t = ||R(t)|| * ||R'(t)||* cos(@) = sqrt(e^2t) * sqrt(2*e^2t) * cos(@)
= sqrt(2*e^4t) * cos(@).

It is easy to then see that @ = cos^-1(e^2t / sqrt(2*e^(4t)), which is obviously still dependant on t.

Any ideas?

2. Jan 24, 2017

### ehild

What is $\sqrt{2e^{4t}}$?

3. Jan 24, 2017

### LCKurtz

Note that $\sqrt{e^{2t}} = e^t$. Then recheck your $\|\vec R'\| = \sqrt{2e^{2t}}$. Get it right and that last line will do it for you.

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text:
## Links forward

### Difficulties with antiderivatives

We saw in the modules on differentiation that many functions can be differentiated. Indeed, knowing how to differentiate elementary functions, and using the product, chain and quotient rules, we can find the derivative of most functions we come across.

So far in this module, we have been able to antidifferentiate every function we encountered. But as it turns out, there are some relatively simple functions for which the antiderivative has no simple formula. Although there are many techniques and tricks for finding antiderivatives, there are no rules as simple as the product, chain and quotient rules for differentiation.

For an interesting and important example, consider the function $$f \colon \mathbb{R}$$ to $$\mathbb{R}$$ defined by $$f(x) = e^{-x^2}$$.

It is easy to differentiate $$f(x)$$ using the chain rule: $$f'(x) = -2xe^{-x^2}$$. However, the antiderivative of $$f(x)$$ has no simple formula. Yet the integral of $$f(x)$$ is very important, because it essentially describes the normal distribution, which is fundamental to probability and statistics.

It is possible to show that the integral of $$f(x)$$, over the whole real line, is

$\int_{-\infty}^{+\infty} e^{-x^2} \; dx = \sqrt{\pi}.$

This may seem to be a surprising result; it is not clear at all where the $$\pi$$ came from.2 Some clever techniques and higher mathematics, which are beyond our scope here, such as area integrals and polar coordinates, can be used to obtain this answer.

Next page - History and applications


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# Is there a notion of hyperbolicity for number rings?

For algebraic curves over a nice enough field $k$, we have a notion of what it means to be hyperbolic: If $\overline{C}$ is a smooth projective curve of genus $g$ and $P_1,\dots,P_n$ are closed points, then the open curve $C=\overline{C}\setminus\{P_1,\dots,P_n\}$ is hyperbolic if $2-2g-\sum \deg P_i<0$.

The general function-field/number-field analogy suggests the following question:

Is there a notion of hyperbolicity for rings of $S$-integers in number fields (possibly along the lines of some inequality appearing in arithmetic intersection theory)?

If there is such a notion of hyperbolicity, are most $S$-integer rings hyperbolic? Is it also true that number rings with trivial or finite étale fundamental group (as in this MO-question) are not hyperbolic?

Here is some of the background which motivated the question: In the function field case, I think I have convinced myself that the slightly stronger inequality $2-2g-n<-1$ implies that there will be cuspidal cohomology for the group $SL_2(k[C])$ (in the sense that the quotient of the product of Bruhat-Tits trees for the points $P_1,\dots,P_n$ is rationally non-contractible). I am trying to understand if similar statements can be made in the number-field situation. Therefore, the ideal answer to the question would be some numerical inequality which relates to Harder's Gauß-Bonnet formula for cohomology of arithmetic groups, implying that hyperbolicity forces non-trivial rational cohomology for the arithmetic group $SL_2(\mathcal{O}_{K,S})$.

• There is an analogy between number fields and 3-manifolds proposed by Barry Mazur where primes in the ring of integers correspond to prime knots, cf. math.harvard.edu/~mazur/papers/alexander_polynomial.pdf The field of rational numbers is supposed to correspond to the 3-sphere. So presumably the field of rational numbers is not hyperbolic. Dec 8, 2016 at 10:51
• Do we ask that an extension (cover) of a hyperbolic number ring should be hyperbolic? Dec 10, 2016 at 19:10

## 1 Answer

My intuition is that there will not be a precise definition of a hyperbolic number field. However, there there may be some number fields you can confidently say are hyperbolic.

Consider the following characterization of a hyperbolic complete curve in the function field case - a degree $d$ cover of $\mathbb P^1$ is hyperbolic if its discriminant $\Delta$ is greater than $q^{2d}$.

This is equivalent to $g>1$ by the Riemann-Hurwitz formula $2-2g = 2d - \log_q (\Delta)$.

So it seems like a number field of degree $d$ should be hyperbolic if the discriminant is greater than $C^{d}$ for some constant $C$. If you want to prove arithmetic analogues of a certain consequence of hyperbolicity, you could try to prove it as a consequence of large discriminant. Is your desired result true for quadratic fields of large discriminant, say?

However for a full definition of hyperbolicity you would need to find a single value of $C$ that is the cutoff for all properties of hyperbolic curves. I don't think a plausible value of $C$ is known - I don't know if one is expected to exist.

One property $C$ should have is that every number field of discriminant $<C^g$, being parabolic, should have finite etale fundamental group. One knows this is true for $C = 22.3$ and false for $C=296.277$ but not a precise optimal value of $C$.

The converse (finite fundamental group implies not hyperbolic) may not actually be true - for many purposes, the etale fundamental group of a number ring is not a perfect analogue of the etale fundamental group of the function field. Instead, to model the etale fundamental group of a number ring, one can use the quotient of the etale fundamental group of a function field by the Frobenius element of a point at $\infty$. The reason is that the decomposition groups at infinite places over number fields are small or even trivial, and so one has to force the same thing to hold in the number field world to get a proper analogue. I'm not sure whether, for function fields, this group is always infinite for hyperbolic curves, but it is certainly true that the pro$-p$ quotients can be finite for hyperbolic curves, whereas even the pro$-p$ quotients of the geometric etale fundamental group are infinite.

• Thanks, that's an interesting suggestion. It is in fact true for imaginary quadratic fields that there are only finitely many where the rational cohomology of $SL_2(\mathcal{O}_{K,S})$ is trivial. Dec 9, 2016 at 9:45
• @MatthiasWendt Cool. I am becoming increasingly convinced that one is supposed to investigate this via a Gauss-Bonnet type formula (or trace formula, or Kusnetsov formula). These should express the dimension of the cuspidal cohomology as a main term (the volume) plus secondary contributions - one from the cusps, one from each element of finite order in $SL_2(\mathcal O_{K})$, and one from the trivial representation. One expresses these all in terms of $\Delta_K$ and then tries to show that, for $\Delta_K$ sufficiently large, the main term dominates the error terms. Dec 9, 2016 at 10:33


text:
Version 27 (modified by MarkelGarcia, 9 years ago) (diff)

--

Reforecast Tutorial

How to get driving model (NCEP) data.

In this example, the publicly available NCEP Reanalysis (run 1) data are going to be used. This data can be dowloaded from http://nomad3.ncep.noaa.gov/pub/reanalysis-1/6hr in GRIB format. These are monthly files that get updated each month nearly in real time. Two files are needed for each month, one with the pressure level data, labeled "pgb", and other one with 2D data, labeled "grb2d". extdata_path defined in experiment.wrf4g must point to the folder where these files are located. Alternatively, it is possible to write a preprocessor that downloads the data itself. Note that the file names must be parsed by the preprocessor. In this case, if both files are located into the same folder, and provided the extension ".grb" is appended to them, the default preprocessor will parse them correctly, since it looks for monthly files with year/month (YYYY/mm) into their names. For example, the files for December 2010 should be:

grb2d201001.grb
pgb.ft00.201001.grb

Creating a WRF experiment

Keeping organized

Before starting to create an experiment, is good practice to create some directories to be tidy. For example, if our project is called "seawind", we can create the following directory hierarchy.

projects/seawind/submit/exp1
projects/seawind/submit/exp2
...
projects/seawind/domains
projects/seawind/data
projects/seawind/scripts
projects/seawind/figures

Of course, many other combinations are possible, depending in the organization of the resources available to the user.

The test experiment

Before creating a large experiment, with many chunks and realizations, it is convenient to run a smaller test experiment with exactly the same model configuration. This way we can see that everything is working as we want. Frequently, some attempts are needed before WRF runs, because of mistakes in the configuration files or in the set up of input files.

Go to the "submit" folder and create another folder called "sw_test":

cd projects/seawind/submit
mkdir sw_test
cd sw_test

Now we need to copy here the templates of experiment.wrf4g and resources.wrf4g.

cp $WRF4G_LOCATION/experiments/wrfuc_single_serial/experiment.wrf4g . cp$WRF4G_LOCATION/etc/resources.wrf4g .

Now we can configure our test experiment, following the instructions in WRF4Gexperiment_wrf4g and WRF4Gresources_wrf4g. Note that, as it is a reforecast, we need to use the multiple dates configuration variables.

Once we are finished, we can use the command line interface (CPI) to prepare and submit our test experiment. Before submitting check in the output of wrf4g_prepare that the chunks and realizations being created are those that we wanted.

wrf4g_prepare
wrf4g_submit -e sw_test

Now we can monitor the experiment using wrf4g_status, along with the "watch" command.

watch wrf4g_status -l -e sw_test

Probably it will fail in the first attempts, so don't worry. If it fails, see how to manage WRF4G errors to look for the error.

Once the test experiment has run successfully, we should check that the output looks fine and that it contains all the variables that we want. Tools like ncview and ncdump are very useful for this task.

The experiment

At this point we are done with the most difficult issues, and we are ready to set up the reforecast experiment itself. Simply create another folder inside "submit" named "sw_ncep", and copy there the configuration files of the test experiment.

cd ..
mkdir sw_ncep
cp -r ../sw_test/* .

Then change the experiment_name and extend the end_date to the end date of the full experiment. In this example we will only run one month (January 2010), but it could be many decades. So our dates would be:

start_date="2009-12-31_18:00:00"
end_date="2010-01-31_06:00:00"

Finally, prepare and submit the experiment with WRF4G CLI, as before.

wrf4g_prepare
wrf4g_submit -e sw_test

Experiment monitoring

With wrf4g_status user can monitor the state of all the realizations of the experiment. If the experiment is large, wrf4g_status can be used in combination with shell tools as grep or awk to filter the list with some criteria. For example:

wrf4g_status -l -e sw_test | grep Finished

Returns all the Finished realizations. Or, more complicated

wrf4g_status -l -e sw_test | grep -v Finished | grep -v Submitted | grep -v Failed

Returns all the realizations that are currently in some stage of the WRF4G workflow (Down. Bin., ungrib, metgrid..., etc.)

If wrf4g_submit is called again, the realizations in Failed status are re-submitted. This is very useful to re-submit simulations after they crashed because of some problems in the computing infrastructure.

Also, some realizations may fail because particular problems. Unfortunately, there are a lot of things that can fail, and covering them in this tutorial is not possible. There is a page in this wiki called WRFKnownProblems where many of them are commented. After years using WRF we still find new error messages sometimes. In a few realizations, WRF may crash because of some points not filling "cfl" criteria. This numerical instabilities arise when too strong gradients do appear for vertical velocity or some other variable. They can be solved using a lower timestep. However, each WRF4G experiment has a fixed timestep defined. Thus, in this case, a new experiment with a lower timestep_dxfactor (e.g. sw_failed_days with timestep_dxfactor = 5) must be created.

Other more complicated situations can occur. For example, if there is a blackout, the jobs can fail before they can send the "Failed" signal to the database. In that case, they can appear to be indefinitely in "WRF" status. These kind of problems need a less confortable monitoring, such as entering to the working nodes and use the commands "top" or "ps -ef" to see it WRF is really running.

When a realization has failed but does not appear with the "Failed" status, it can be resubmitted using the --force flag of wrf4g_submit. Note also that wrf4g_submit can be used refering only to one realization or chunk, using the flags, -r or -c.

The monitoring and maintenance of an experiment can be shared my different users, provided they connect their WRF4G to the same database.


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# Math formulas API¶

## Why we need a formulas API¶

Their is a few reasons why having a library of mathematical formula for theano is a good reason:

• Some formula have some special thing needed for the gpu.
• Sometimes we need to cast to floatX...
• Some formula have numerical stability problem.

• Some formula gradiant have numerical stability problem. (Happen more frequently then the previous ones)
• If theano don’t always do some stability optimization, we could do it manually in the formulas
• Some formula as complex to implement and take many try to do correctly.

• Can mimic the hierarchy of other library to ease the migration to theano

Having a library help in that we solve those problem only once.

## What is a formula¶

We define formulas as something that don’t have a state. They are implemented as python function that take theano variable as input and they output theano variable. If you want state, look at what the others commities will do.

## Formulas documentation¶

We must respect what the coding commitee have set for the docstring of the file and of the function.

• A latex mathematical description of the formulas(for picture representation in generated documentation)

• Tags(for searching):
• a list of lower level fct used
• category(name of the submodule itself)
• Tell if we did some work to make it more numerical stable. Do theano do the optimization needed?

• Tell if the grad is numericaly stable? Do theano do the optimization needed?

• Tell if work/don’t/unknow on gpu.

• Tell alternate name

• Tell the domaine, range of the input/output(range should use the english notation of including or excluding)

## Proposed hierarchy¶

Here is the proposed hierarchy for formulas:

• pylearn.formulas.costs: generic / common cost functions, e.g. various cross-entropies, squared error, abs. error, various sparsity penalties (L1, Student)
• pylearn.formulas.regularization: formulas for regularization
• pylearn.formulas.linear: formulas for linear classifier, linear regression, factor analysis, PCA
• pylearn.formulas.nnet: formulas for building layers of various kinds, various activation functions, layers which could be plugged with various costs & penalties, and stacked
• pylearn.formulas.ae: formulas for auto-encoders and denoising auto-encoder variants
• pylearn.formulas.noise: formulas for corruption processes
• pylearn.formulas.rbm: energies, free energies, conditional distributions, Gibbs sampling
• pylearn.formulas.trees: formulas for decision trees
• pylearn.formulas.boosting: formulas for boosting variants
• pylearn.formulas.maths for other math formulas
• pylearn.formulas.scipy.stats: example to implement the same interface as existing lib

etc.

## Example¶

"""
This script defines a few often used cost functions.
"""
import theano
import theano.tensor as T
from tags import tags

@tags('cost','binary','cross-entropy')
def binary_crossentropy(output, target):
""" Compute the crossentropy of binary output wrt binary target.

.. math::
L_{CE} \equiv t\log(o) + (1-t)\log(1-o)

:type output: Theano variable
:param output: Binary output or prediction :math:\in[0,1]
:type target: Theano variable
:param target: Binary target usually :math:\in\{0,1\}
"""
return -(target * tensor.log(output) + (1.0 - target) * tensor.log(1.0 - output))


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## First Moon observation with my QO-100 station

A month ago, I spoke about planning the passes of the Moon through the beam of my QO-100 station. These give an occasion to observe the Moon without moving a dish that is pointing to Es’hail 2. The next opportunity after writing that post was on May 16 at 20:16 UTC.

Since I wasn’t going to be at home at that time, I programmed my computer to make a recording for later analysis. I recorded 4MHz of spectrum centred at 10367.5MHz using a LimeSDR connected to the LNB that I use to receive QO-100. The recording was planned to be 30 minutes long starting at 20:01 UTC, but for some reason only approximately 27 minutes were recorded.

This kind of events can be used to measure Moon noise and receive 10GHz EME signals. This post is an analysis of my recording, looking at these two things.

Continue reading “First Moon observation with my QO-100 station”

The Amateur transponders of Es’hail 2 have their downlink in the 10GHz Amateur band. Even though the path to the satellite through the atmosphere is rather short, in extreme weather conditions it is possible to observe a small amount of fading in the signal. Two days ago there was intense rain over Madrid. As I’m often recording the power of the narrowband transponder beacons and the transponder noise floor, I have examined my data to see if the effect of the rain is visible.

The data is plotted in the figure below. See this post for an explanation of the measurements.

The power of the beacons is not very stable. It can vary up to 2 or 3dB along the course of the day. Therefore, it is not so easy to measure the drop in signal power caused by rain. However, it is noticeable that on April 24, between 05:00 and 17:00 UTC, the power of the beacons varies much more rapidly than usually. A small ripple of 0.5dB of amplitude is visible on the data. I think that this ripple is caused by varying rain intensity. Therefore, the data seems to suggest that the rains two days ago caused up to 0.5dB of fading in the signal.

As seen from my station, the satellite is at an elevation of $$\theta = 33.6^\circ$$, so assuming a slant factor of $$1/\sin \theta = 1.8$$, so taking a typical height of around 1km for the column of rain (see the corresponding METAR for Madrid airport), we get an attenuation on the order of 0.3dB/km. However, all the measurements used here are too imprecise to obtain any good conclusions. See this related post, in which I measured a 2.5dB increase in the noise floor at 12GHz during a hailstorm, but no change in signal power.

## QO-100 beacons power

In the QO-100 (Es’hail 2) narrow band transponder, the recommendation for the adjustment of your downlink signal power is not to be stronger than the beacon. This was also the recommended usage of the old AO-40. Since the transponder has two beacons marking the transponder edges: a CW beacon marking the lower edge and a 400baud BPSK beacon marking the upper edge, there has been some debate on Twitter about which beacon does this recommendation refer to and what does “stronger” mean.

Of course, more formally, signal strength means power, which is a well defined physical concept, so there should be no argument about what does power mean. However, there are two different power measurements used for RF: average power and peak envelope power. I will assume that the recommendation refers to average power, not to peak envelope power. This makes more sense from the point of view of the power budget of the satellite amplifier (The total average power it needs to deliver is just the sum of the average powers of the signals of all the users, while the behaviour of the peak envelope power is much more complicated).

Also, I think that using peak envelope power for this restriction would be a very strict requirement on high PAPR signals. Note that the PAPR of CW is 0dB and the PAPR of BPSK is between 2 and 3dB, depending on the pulse shaping, so these are rather low PAPRs. For comparison, a moderately compressed SSB voice signal has a PAPR of 6dB.

In my opinion, the main problem with these discussions about “signal strength” is that many people are trying to judge power by looking at their waterfall or spectrum display and seeing what signal looks “higher”. This kind of measurement is not any good, because it doesn’t take signal bandwidth into account, depends on the FFT size, the window function, etc. It doesn’t help that many popular SDR software don’t have a good signal meter displaying the average power of the signal tuned in the passband.

In any case, I was curious about whether the power of the two beacons is the same and whether there is any interesting change over time. I have made a GNU Radio flowgraph that measures the power of each of the two beacons and of the transponder noise, and saves them to a file for later analysis.

## Decoding the QO-100 beacon with gr-satellites

On February 14, the Amateur transponders on Es’hail 2 (which now has the AMSAT designation QO-100) were inaugurated. Since then, two beacons are being transmitted by the groundstation in Doha (Qatar) through the narrowband transponder. These beacons mark the edges of the transponder.

The lower beacon is CW, while the upper beacon is a 400baud BPSK beacon that uses the same format as the uncoded beacon of AO-40. I have already talked about the AO-40 uncoded beacon in an older post, including the technical details.

Based on my AO-40 decoder in gr-satellites, I have made a decoder for the QO-100 beacon. Patrick Dohmen DL4PD has been kind enough to write some instructions about how to use the old ao40_uncoded decoder with the BATC WebSDR. I recommend that you use the new qo100 decoder. You just have substitute ao40_uncoded by qo100 in Patrick’s instructions

As additional hints, I can say that for the best decoding, the beacon must be centred at 1.5kHz into the SSB passband. The centre of the signal is easy to spot because there is a null at the centre, due to the use of Manchester encoding. Frequency stability is somewhat important with this decoder, so if your LNB drifts too much you may run into problems.

The SNR of the beacon over the transponder noise floor is rather high, so you should achieve a clean decoding unless you are using a very small station and you have the transponder noise way below your receiver noise floor.

The following data is being currently transmitted on the beacon (the timestamps and packet numbers are added by gr-satellites):

2019-02-19 21:56:27
Packet number 68
K HI de QO-100 (DL50AMSAT BOCHUM
UPT: 3d 0h 29m CMD: 91 LEI_REQ: 0 LEI_ACT: 0
TEMP: 56 C VOLTAGES: 1.0 1.8 1.0 1.0 1.8 1.5 1.3 0.0 0.5 Volts
TFL: 0 TFE: 0 TFH: 0 HFF: 0 HTH: 0 HR: 0

2019-02-19 21:56:53
Packet number 69
L HI de QO-100 (DL50AMSAT BOCHUM
EXPERIMENTAL MODE. Measurements and tests being conducted,
experimental transponder use OK, but expect ground station tests
Watch this space and www.amsat-dl.org for further announcements

## In-orbit testing of Es’hail 2 Amateur transponders

Yesterday, December 23, MELCO carried out some in-orbit tests of the Es’hail 2 Amateur radio transponders. Since Es’hail 2 is currently under commissioning, it was expected that at some point the Amateur transponders would be activated for testing, but no announcement of the tests was done in advance. At around 11:00 UTC, Rob Janssen PE1CHL, noticed that the narrowband transponder was active and a carrier signal was being transmitted through it.

Since then, I monitored most of the tests and sent updates on Twitter, together with other people (see also the posts in the AMSAT-DL forum). Without knowing the details of the test plans, we limited ourselves to watching and following the tests that were being made. If some schedule of the tests had been published in advanced, we could have thought, prepared and performed some interesting measurements on the downlink signals.

I understand that since these tests are carried out by MELCO, AMSAT-DL might not have the specific details, but still I think that AMSAT-DL is publishing very little information about Es’hail 2 events. It was only at 22:35 UTC that AMSAT-DL published a small note on Twitter about the tests. I think the greatest concern is that people start transmitting through the transponder, interfering with the tests. However, since news spread very fast these days through social media, I think that publishing more information rather than keeping things discreetly serves better to prevent people from using the transponder during the commissioning. In any case, I’ll repeat it here:

Es’hail 2 is currently under commissioning. The 2.4GHz uplink of the Amateur transponders should never be used until authorized by AMSAT-DL. The Amateur transponders will sometimes be enabled for in-orbit testing by the MELCO/Es’hailSat/AMSAT-DL engineers. Relax, sit back, and watch the tests on the 10GHz downlink.

I also think that publishing more information would be beneficial to educate the community of radio Amateurs. Some people have asked me about the concept of in-orbit tests. After a satellite is launched into orbit, the performance of all its systems is tested to ensure that it matches design specifications, simulations, and pre-launch tests done on ground. This is important to guarantee that any problems, malfunction or damage that occurred during the launch can be diagnosed and hopefully mitigated by activating backup systems or other reliability measures. In-orbit testing of large satellites can take several months, since there are many complex systems that need to be tested remotely.

In the case of the Amateur radio payload of Es’hail 2, MELCO is carrying out the tests, since the payload was built by MELCO according to the design specifications by AMSAT-DL. The kind of tests they are performing are related to the performance of the bent-pipe transponders. They sweep in frequency the transponders to make sure that the passband shape is as expected. They transmit carriers of different power levels to check for linearity of the transponder and AGC performance, and they try different gain/power level settings of the transponder power amplifier to make sure it performs correctly over all its working range.

This is a rough account of the tests that were made yesterday, using my tweets as a sort of activity log.

Continue reading “In-orbit testing of Es’hail 2 Amateur transponders”

## Recommendations for receiving Es’hail 2

A couple days ago, Janos Tolgyesi HG5APZ asked me by email about different hardware setups to receive the Amateur radio transponders on Es’hail 2, with an interest on inexpensive but effective solutions. He was quite happy with my detailed reply and convinced me to turn it into a blog post, so that other people can learn from it.

This post is intended for people that do not know much about Es’hail 2 but are interested in receiving it. If you’ve been investigating about the different setups that people are doing to receive it, then probably you’ll not learn anything new here. The post addresses questions such as “do I need a modified LNB” and similar.

Continue reading “Recommendations for receiving Es’hail 2”

## Aircraft reflections of a 2.3GHz beacon

A couple months ago, Andrés Calleja EB4FJV installed a 2.3GHz beacon in his home in Colmenar Viejo, Madrid. The beacon has 2W of power, radiates with an omnidirectional antenna in the vertical polarization, and transmits a tone and CW identification at the frequency 2320.865MHz.

Since Colmenar Viejo is only 10km away from Tres Cantos, where I live, I can receive the beacon with a very strong signal from home. The Madrid-Barajas airport is also quite near (15km to the threshold of runway 18R) and several departure and approach aircraft routes pass nearby, particularly those flying over the Colmenar VOR. Therefore, it is quite easy to see reflections off aircraft when listening to the beacon.

On July 8 I did a recording of the beacon from 10:04 to 11:03 UTC from the countryside just outside Tres Cantos. In this post I will examine the aircraft reflections seen in the recording and match them with ADS-B aircraft position and velocity data obtained from adsbexchange.com. This will show the locations and trajectories which produce reflections strong enough to be detected.

## DSLWP-B’s journey to the Moon: part III

This is a follow-up on the series about DSLWP-B’s orbital dynamics (see part I and part II). In part I we looked at the tracking files published by Wei Mingchuan BG2BHC, which list the position and velocity of the satellite in ECEF coordinates, and presented basic orbit propagation with GMAT. In part II we explored GMAT’s capabilities to plan and perform manoeuvres, making a tentative simulation of DSLWP-B’s mid-course correction and lunar orbit injection. Now we turn to the study of DSLWP-B’s elliptical lunar orbit.

In this post we will examine the Keplerian elements of the orbits described by each of the tracking files published so far. We will also use Scott Tilley VE7TIL’s Doppler measurements of the S-band beacon of DSLWP-B to validate and determine the orbit.

## Update on phase noise of 27MHz references

This is a follow up to a previous post where I investigated the phase noise of 27MHz references to be used for a 10GHz receiver. Dieter DF9NP has being kind enough to send me a 10MHz 0.25ppm TCXO to do some more tests.

I’ve connected the 10MHz TCXO to the DF9NP 27MHz PLL and used it to receive the beacon of BADR-5, as I did in the previous post. The phase noise of the 10MHz TCXO + 27MHz PLL can be seen in the following figure.

For comparison, see below the phase noise with the DF9NP 10MHz GPSDO and 27MHz PLL. There is not much difference between both. This seems to indicate that the culprit of the phase noise is the 27MHz PLL, as the 10MHz TCXO should be quite clean.

## Phase noise of a Baofeng UV-5R 10GHz signal

Several of the Baofeng chinese handheld radios generate a weak 10GHz signal while in receive mode. Thus, they are a popular cheap and quick 10GHz signal source for tests. To generate a 10GHz signal, you have to tune the Baofeng to the 70cm band (for instance, 432MHz). The radio will generate a weak 24th harmonic while in receive mode. If you want a steady carrier, you have to set the squelch to zero. Otherwise you will just get beeps as the radio wakes up periodically to check for a signal. Lately, I’ve being investigating phase noise and reciprocal mixing of 10GHz receiver systems. A natural question is how good is the phase noise of a Baofeng used as a 10GHz signal source and whether it can be used to check if the phase noise performance of a receiver is acceptable. It turns out that it is not so noisy as one may first think.


text:
## Perry, Peter A.

Compute Distance To:
 Author ID: perry.peter-a Published as: Perry, Peter A.; Perry, Peter; Perry, P.; Perry, P. A. Homepage: http://www.ms.uky.edu/~perry/ External Links: MGP · ResearchGate
 Documents Indexed: 78 Publications since 1980, including 1 Book 5 Contributions as Editor Co-Authors: 53 Co-Authors with 61 Joint Publications 1,736 Co-Co-Authors
all top 5

### Co-Authors

 22 single-authored 9 Brooks, Robert Wolfe 7 Sulem, Catherine 6 Liu, Jiaqi 5 Borthwick, David 5 Hislop, Peter D. 5 Petersen, Peter V 4 Brown, Russell Murray 4 Gordon, Carolyn S. 4 Hryniv, Rostyslav O. 4 Mykytyuk, Yaroslav V. 3 Hagedorn, George A. 3 Music, Michael 3 Siltanen, Samuli 3 Simon, Barry 2 Deift, Percy A. 2 Froese, Richard 2 Galvez, Cherie 2 Gesztesy, Fritz 2 Jensen, Arne 2 Judge, Christopher M. 2 Kappeler, Thomas 2 McGowan, Jeffrey 2 Schlag, Wilhelm 2 Schüth, Dorothee 2 Shen, Zhongwei 2 Shubin, Mikhail Aleksandrovich 2 Sigal, Israel Michael 2 Tang, Siuhung 2 Topalov, Peter J. 1 Albeverio, Sergio A. 1 Barnett, Alex H. 1 Caudill, Lester F. jun. 1 Christiansen, Tanya J. 1 Croke, Ryan 1 Cycon, Hans L. 1 Dodziuk, Jozef 1 Ehrenborg, Richard 1 Epstein, Charles Lawrence 1 Evans, William Desmond 1 Gornet, Ruth 1 Kitaev, Sergey 1 Lytle, George 1 Miller, Peter D. 1 Mourre, Éric 1 Mueller, Jennifer L. 1 Ott, Katharine A. 1 Patterson, Samuel James 1 Saut, Jean-Claude 1 Schueller, Albert W. 1 Siedentop, Heinz Karl Heinrich 1 Stahel, Andreas 1 Uribe, Alejandro 1 Williams, Floyd L. 1 Yang, Paul C. P.
all top 5

### Serials

 5 Communications in Mathematical Physics 4 Duke Mathematical Journal 4 Journal of Functional Analysis 4 The Journal of Geometric Analysis 4 Communications in Partial Differential Equations 3 Inverse Problems 3 Journal für die Reine und Angewandte Mathematik 3 IMRN. International Mathematics Research Notices 3 Proceedings of Symposia in Pure Mathematics 3 Analysis & PDE 2 Nonlinearity 2 Commentarii Mathematici Helvetici 2 Geometric and Functional Analysis. GAFA 2 Bulletin of the American Mathematical Society. New Series 1 Communications on Pure and Applied Mathematics 1 Acta Mathematica 1 Annales Academiae Scientiarum Fennicae. Series A I. Mathematica 1 Geometriae Dedicata 1 Indiana University Mathematics Journal 1 Inventiones Mathematicae 1 Mathematische Zeitschrift 1 Quarterly of Applied Mathematics 1 Transactions of the American Mathematical Society 1 Journal of Operator Theory 1 Annales de l’Institut Henri Poincaré. Analyse Non Linéaire 1 SIAM Journal on Applied Mathematics 1 SIAM Journal on Mathematical Analysis 1 Annales de l’Institut Henri Poincaré. Physique Théorique 1 Journées Équations aux Dérivées Partielles (Saint-Jean-de-Monts) 1 Mathematical Research Letters 1 Annals of Mathematics. Second Series 1 Comptes Rendus. Mathématique. Académie des Sciences, Paris 1 International Journal of Pure and Applied Mathematics 1 Contemporary Mathematics 1 Fields Institute Communications 1 Mathematical Reports (Chur) 1 Journal of Spectral Theory 1 Journal of Combinatorics
all top 5

### Fields

 48 Partial differential equations (35-XX) 36 Global analysis, analysis on manifolds (58-XX) 19 Operator theory (47-XX) 18 Dynamical systems and ergodic theory (37-XX) 16 Quantum theory (81-XX) 13 Number theory (11-XX) 10 Differential geometry (53-XX) 6 Ordinary differential equations (34-XX) 5 General and overarching topics; collections (00-XX) 5 Functions of a complex variable (30-XX) 4 Optics, electromagnetic theory (78-XX) 3 Numerical analysis (65-XX) 2 Combinatorics (05-XX) 2 Topological groups, Lie groups (22-XX) 2 Several complex variables and analytic spaces (32-XX) 2 Manifolds and cell complexes (57-XX) 2 Fluid mechanics (76-XX) 1 Group theory and generalizations (20-XX) 1 Special functions (33-XX) 1 Harmonic analysis on Euclidean spaces (42-XX) 1 Abstract harmonic analysis (43-XX) 1 Integral equations (45-XX) 1 Functional analysis (46-XX) 1 Calculus of variations and optimal control; optimization (49-XX) 1 Probability theory and stochastic processes (60-XX) 1 Statistical mechanics, structure of matter (82-XX) 1 Relativity and gravitational theory (83-XX)

### Citations contained in zbMATH Open

67 Publications have been cited 791 times in 575 Documents Cited by Year
Spectral analysis of N-body Schrödinger operators. Zbl 0477.35069
Perry, P.; Sigal, I. M.; Simon, B.
1981
Multiple commutator estimates and resolvent smoothness in quantum scattering theory. Zbl 0561.47007
Jensen, Arne; Mourre, Éric; Perry, Peter
1984
The divisor of Selberg’s zeta function for Kleinian groups. Appendix A by Charles Epstein. Zbl 1012.11083
Patterson, S. J.; Perry, Peter A.
2001
Scattering theory by the Enss method. Ed. by B. Simon. Zbl 0529.35004
Perry, Peter A.
1983
The spectrum of relativistic one-electron atoms according to Bethe and Salpeter. Zbl 0857.47041
Evans, William Desmond; Perry, Peter; Siedentop, Heinz
1996
Global existence for the derivative nonlinear Schrödinger equation by the method of inverse scattering. Zbl 1358.35173
Liu, Jiaqi; Perry, Peter A.; Sulem, Catherine
2016
The Laplace operator on a hyperbolic manifold. II: Eisenstein series and the scattering matrix. Zbl 0677.58044
Perry, Peter A.
1989
Long-time behavior of solutions to the derivative nonlinear Schrödinger equation for soliton-free initial data. (Comportement aux temps longs des solutions de l’équation de Schrödinger nonlinéraire avec dérivée en l’absence de solitons.) Zbl 1382.35271
Liu, Jiaqi; Perry, Peter A.; Sulem, Catherine
2018
Isospectral sets of conformally equivalent metrics. Zbl 0667.53037
Brooks, Robert; Perry, Peter; Yang, Paul
1989
Soliton resolution for the derivative nonlinear Schrödinger equation. Zbl 1408.35175
Jenkins, Robert; Liu, Jiaqi; Perry, Peter; Sulem, Catherine
2018
On the dimension of the attractor for the non-homogeneous Navier-Stokes equations in non-smooth domains. Zbl 0969.35105
Brown, Russell M.; Perry, Peter A.; Shen, Zhongwei
2000
Global well-posedness and long-time asymptotics for the defocussing Davey-Stewartson II equation in $$H^{1,1}(\mathbb{C})$$. Zbl 1354.37067
Perry, Peter A.
2016
Scattering poles for asymptotically hyperbolic manifolds. Zbl 1009.58021
Borthwick, David; Perry, Peter
2002
Propagation of states in dilation analytic potentials and asymptotic completeness. Zbl 0471.47007
Perry, Peter A.
1981
Selberg’s zeta function and the spectral geometry of geometrically finite hyperbolic surfaces. Zbl 1079.58023
Borthwick, David; Judge, Chris; Perry, Peter A.
2005
The Miura map on the line. Zbl 1089.35058
Kappeler, Thomas; Perry, Peter; Shubin, Mikhail; Topalov, Peter
2005
Mellin transforms and scattering theory I. Short range potentials. Zbl 0445.47009
Perry, Peter A.
1980
The Laplace operator on a hyperbolic manifold. I: Spectral and scattering theory. Zbl 0631.58030
Perry, Peter A.
1987
Commutator methods and Besov space estimates for Schrödinger operators. Zbl 0574.35022
Jensen, Arne; Perry, Peter
1985
A spectral approach to consecutive pattern-avoiding permutations. Zbl 1247.05013
Ehrenborg, Richard; Kitaev, Sergey; Perry, Peter
2011
Compactness and finiteness theorems for isospectral manifolds. Zbl 0737.53038
Brooks, Robert; Perry, Peter; Petersen, Peter V
1992
CR-invariants and the scattering operator for complex manifolds with boundary. Zbl 1159.32023
Hislop, Peter D.; Perry, Peter A.; Tang, Siu-Hung
2008
Isospectral and isoscattering manifolds: a survey of techniques and examples. Zbl 1087.58021
Gordon, Carolyn; Perry, Peter; Schueth, Dorothee
2005
The Selberg zeta function and a local trace formula for Kleinian groups. Zbl 0697.10027
Perry, Peter A.
1990
Exceptional circles of radial potentials. Zbl 1276.78001
Music, M.; Perry, P.; Siltanen, S.
2013
Miura maps and inverse scattering for the Novikov-Veselov equation. Zbl 1405.37081
Perry, Peter
2014
Isospectral sets for fourth-order ordinary differential operators. Zbl 0912.34068
Caudill, Lester F. jun.; Perry, Peter A.; Schueller, Albert W.
1998
Determinants of Laplacians and isopolar metrics on surfaces of infinite area. Zbl 1040.58013
Borthwick, David; Judge, Chris; Perry, Peter A.
2003
A Poisson summation formula and lower bounds for resonances in hyperbolic manifolds. Zbl 1035.58020
Perry, Peter A.
2003
A Mourre estimate and related bounds for hyperbolic manifolds with cusps of non-maximal rank. Zbl 0729.58051
Froese, R.; Hislop, P.; Perry, P.
1991
Selberg zeta function and trace formula for the BTZ black hole. Zbl 1056.11030
Perry, Peter A.; Williams, Floyd L.
2003
Inverse scattering for Schrödinger operators with Miura potentials. I: Unique Riccati representatives and ZS-AKNS systems. Zbl 1181.35330
Frayer, C.; Hryniv, R. O.; Mykytyuk, Ya V.; Perry, P. A.
2009
Global well-posedness for the derivative non-linear Schrödinger equation. Zbl 1412.35309
Jenkins, Robert; Liu, Jiaqi; Perry, Peter A.; Sulem, Catherine
2018
Exponential bounds and semi-finiteness of point spectrum for N-body Schrödinger operators. Zbl 0546.35051
Perry, Peter A.
1984
Isoscattering Schottky manifolds. Zbl 0966.58018
Brooks, Robert; Gornet, Ruth; Perry, Peter
2000
Asymptotics of the length spectrum for hyperbolic manifolds of infinite volume. Zbl 0986.11059
Perry, P. A.
2001
On Cheeger’s inequality. Zbl 0809.53049
Brooks, Robert; Perry, Peter; Petersen, Peter V
1993
The Laplace operator on hyperbolic three manifolds with cusps of non- maximal rank. Zbl 0763.58028
Froese, R.; Hislop, P.; Perry, P.
1991
The Novikov-Veselov equation: theory and computation. Zbl 1330.35375
Croke, R.; Mueller, J. L.; Music, M.; Perry, P.; Siltanen, S.; Stahel, A.
2015
Solutions of mKdV in classes of functions unbounded at infinity. Zbl 1155.35087
Kappeler, T.; Perry, P.; Shubin, M.; Topalov, P.
2008
Local time decay of high energy scattering states for the Schrödinger equation. Zbl 0538.35026
Cycon, Hans L.; Perry, Peter A.
1985
Inverse scattering results for manifolds hyperbolic near infinity. Zbl 1229.58024
Borthwick, David; Perry, Peter A.
2011
The derivative nonlinear Schrödinger equation: global well-posedness and soliton resolution. Zbl 1434.35180
Jenkins, Robert; Liu, Jiaqi; Perry, Peter; Sulem, Catherine
2020
Spectral theory and mathematical physics. A festschrift in honor of Barry Simon’s 60th birthday. Ergodic Schrödinger operators, singular spectrum, orthogonal polynomials, and inverse spectral theory. Based on the SimonFest conference, Pasadena, CA, USA, March 27–31, 2006. Zbl 1110.00015
2007
The spectral geometry of geometrically finite hyperbolic manifolds. Zbl 1138.58009
Perry, Peter
2007
Asymptotic completeness for certain three-body Schrödinger operators. Zbl 0506.35026
Hagedorn, George A.; Perry, Peter A.
1983
A trace-class rigidity theorem for Kleinian groups. Zbl 0861.57017
Perry, Peter A.
1995
Inverse scattering for Schrödinger operators with Miura potentials. II: Different Riccati representatives. Zbl 1232.34113
Hryniv, Rostyslav O.; Mykytyuk, Yaroslav V.; Perry, Peter A.
2011
Isophasal scattering manifolds in two dimensions. Zbl 1005.58011
Brooks, Robert; Perry, Peter A.
2001
Action of a scattering map on weighted Sobolev spaces in the plane. Zbl 1375.35287
Brown, R. M.; Ott, K. A.; Perry, P. A.
2016
Global existence for the derivative nonlinear Schrödinger equation with arbitrary spectral singularities. Zbl 1451.35186
Jenkins, Robert; Liu, Jiaqi; Perry, Peter; Sulem, Catherine
2020
The Selberg zeta function and scattering poles for Kleinian groups. Zbl 0723.11028
Perry, Peter A.
1991
Continuous families of isophasal scattering manifolds. Zbl 1139.58022
Gordon, Carolyn; Perry, Peter
2006
A remark on continuum eigenfunctions of N-body Schrödinger operators. Zbl 0576.35107
Perry, Peter A.
1984
Asymptotic completeness of certain four-body Schrödinger operators. Zbl 0589.35032
Hagedorn, George A.; Perry, Peter A.
1986
Sobolev mapping properties of the scattering transform for the Schrödinger equation. Zbl 1221.34230
Hryniv, Rostyslav O.; Mykytyuk, Yaroslav V.; Perry, Peter A.
2011
Resonances for manifolds hyperbolic near infinity: optimal lower bounds on order of growth. Zbl 1228.53045
Borthwick, D.; Christiansen, T. J.; Hislop, P. D.; Perry, P. A.
2011
The additive turbulent decomposition for the two-dimensional incompressible Navier-Stokes equations: Convergence theorems and error estimates. Zbl 0922.35123
Brown, Russell M.; Perry, Peter; Shen, Zhongwei
1999
Meromorphic continuation of the resolvent for Kleinian groups. Zbl 0942.58033
Perry, Peter
1999
The Selberg zeta function and scattering poles for Kleinian groups. Zbl 0824.11037
Perry, Peter A.
1995
Closed geodesics in homology classes for convex co-compact hyperbolic manifolds. Zbl 1122.11307
McGowan, Jeffrey; Perry, Peter
2002
Global solutions for the zero-energy Novikov-Veselov equation by inverse scattering. Zbl 1396.37074
Music, Michael; Perry, Peter
2018
Carnot geometry and the resolvent of the sub-Laplacian for the Heisenberg group. Zbl 1035.35027
Perry, Peter A.
2003
Harmonic representatives for cuspidal cohomology classes. Zbl 1250.30037
Dodziuk, Józef; McGowan, Jeffrey; Perry, Peter
2012
Spectral geometry. Based on the international conference, Dartmouth, NH, USA, July 19–23, 2010. Zbl 1253.58001
2012
CR-invariants and the scattering operator for complex manifolds with CR-boundary. Zbl 1096.58009
Hislop, Peter D.; Perry, Peter A.; Tang, Siu-Hung
2006
Nachman’s reconstruction method for the Calderón problem with discontinuous conductivities. Zbl 1437.35356
Lytle, George; Perry, Peter; Siltanen, Samuli
2020
The derivative nonlinear Schrödinger equation: global well-posedness and soliton resolution. Zbl 1434.35180
Jenkins, Robert; Liu, Jiaqi; Perry, Peter; Sulem, Catherine
2020
Global existence for the derivative nonlinear Schrödinger equation with arbitrary spectral singularities. Zbl 1451.35186
Jenkins, Robert; Liu, Jiaqi; Perry, Peter; Sulem, Catherine
2020
Nachman’s reconstruction method for the Calderón problem with discontinuous conductivities. Zbl 1437.35356
Lytle, George; Perry, Peter; Siltanen, Samuli
2020
Long-time behavior of solutions to the derivative nonlinear Schrödinger equation for soliton-free initial data. (Comportement aux temps longs des solutions de l’équation de Schrödinger nonlinéraire avec dérivée en l’absence de solitons.) Zbl 1382.35271
Liu, Jiaqi; Perry, Peter A.; Sulem, Catherine
2018
Soliton resolution for the derivative nonlinear Schrödinger equation. Zbl 1408.35175
Jenkins, Robert; Liu, Jiaqi; Perry, Peter; Sulem, Catherine
2018
Global well-posedness for the derivative non-linear Schrödinger equation. Zbl 1412.35309
Jenkins, Robert; Liu, Jiaqi; Perry, Peter A.; Sulem, Catherine
2018
Global solutions for the zero-energy Novikov-Veselov equation by inverse scattering. Zbl 1396.37074
Music, Michael; Perry, Peter
2018
Global existence for the derivative nonlinear Schrödinger equation by the method of inverse scattering. Zbl 1358.35173
Liu, Jiaqi; Perry, Peter A.; Sulem, Catherine
2016
Global well-posedness and long-time asymptotics for the defocussing Davey-Stewartson II equation in $$H^{1,1}(\mathbb{C})$$. Zbl 1354.37067
Perry, Peter A.
2016
Action of a scattering map on weighted Sobolev spaces in the plane. Zbl 1375.35287
Brown, R. M.; Ott, K. A.; Perry, P. A.
2016
The Novikov-Veselov equation: theory and computation. Zbl 1330.35375
Croke, R.; Mueller, J. L.; Music, M.; Perry, P.; Siltanen, S.; Stahel, A.
2015
Miura maps and inverse scattering for the Novikov-Veselov equation. Zbl 1405.37081
Perry, Peter
2014
Exceptional circles of radial potentials. Zbl 1276.78001
Music, M.; Perry, P.; Siltanen, S.
2013
Harmonic representatives for cuspidal cohomology classes. Zbl 1250.30037
Dodziuk, Józef; McGowan, Jeffrey; Perry, Peter
2012
Spectral geometry. Based on the international conference, Dartmouth, NH, USA, July 19–23, 2010. Zbl 1253.58001
2012
A spectral approach to consecutive pattern-avoiding permutations. Zbl 1247.05013
Ehrenborg, Richard; Kitaev, Sergey; Perry, Peter
2011
Inverse scattering results for manifolds hyperbolic near infinity. Zbl 1229.58024
Borthwick, David; Perry, Peter A.
2011
Inverse scattering for Schrödinger operators with Miura potentials. II: Different Riccati representatives. Zbl 1232.34113
Hryniv, Rostyslav O.; Mykytyuk, Yaroslav V.; Perry, Peter A.
2011
Sobolev mapping properties of the scattering transform for the Schrödinger equation. Zbl 1221.34230
Hryniv, Rostyslav O.; Mykytyuk, Yaroslav V.; Perry, Peter A.
2011
Resonances for manifolds hyperbolic near infinity: optimal lower bounds on order of growth. Zbl 1228.53045
Borthwick, D.; Christiansen, T. J.; Hislop, P. D.; Perry, P. A.
2011
Inverse scattering for Schrödinger operators with Miura potentials. I: Unique Riccati representatives and ZS-AKNS systems. Zbl 1181.35330
Frayer, C.; Hryniv, R. O.; Mykytyuk, Ya V.; Perry, P. A.
2009
CR-invariants and the scattering operator for complex manifolds with boundary. Zbl 1159.32023
Hislop, Peter D.; Perry, Peter A.; Tang, Siu-Hung
2008
Solutions of mKdV in classes of functions unbounded at infinity. Zbl 1155.35087
Kappeler, T.; Perry, P.; Shubin, M.; Topalov, P.
2008
Spectral theory and mathematical physics. A festschrift in honor of Barry Simon’s 60th birthday. Ergodic Schrödinger operators, singular spectrum, orthogonal polynomials, and inverse spectral theory. Based on the SimonFest conference, Pasadena, CA, USA, March 27–31, 2006. Zbl 1110.00015
2007
The spectral geometry of geometrically finite hyperbolic manifolds. Zbl 1138.58009
Perry, Peter
2007
Continuous families of isophasal scattering manifolds. Zbl 1139.58022
Gordon, Carolyn; Perry, Peter
2006
CR-invariants and the scattering operator for complex manifolds with CR-boundary. Zbl 1096.58009
Hislop, Peter D.; Perry, Peter A.; Tang, Siu-Hung
2006
Selberg’s zeta function and the spectral geometry of geometrically finite hyperbolic surfaces. Zbl 1079.58023
Borthwick, David; Judge, Chris; Perry, Peter A.
2005
The Miura map on the line. Zbl 1089.35058
Kappeler, Thomas; Perry, Peter; Shubin, Mikhail; Topalov, Peter
2005
Isospectral and isoscattering manifolds: a survey of techniques and examples. Zbl 1087.58021
Gordon, Carolyn; Perry, Peter; Schueth, Dorothee
2005
Determinants of Laplacians and isopolar metrics on surfaces of infinite area. Zbl 1040.58013
Borthwick, David; Judge, Chris; Perry, Peter A.
2003
A Poisson summation formula and lower bounds for resonances in hyperbolic manifolds. Zbl 1035.58020
Perry, Peter A.
2003
Selberg zeta function and trace formula for the BTZ black hole. Zbl 1056.11030
Perry, Peter A.; Williams, Floyd L.
2003
Carnot geometry and the resolvent of the sub-Laplacian for the Heisenberg group. Zbl 1035.35027
Perry, Peter A.
2003
Scattering poles for asymptotically hyperbolic manifolds. Zbl 1009.58021
Borthwick, David; Perry, Peter
2002
Closed geodesics in homology classes for convex co-compact hyperbolic manifolds. Zbl 1122.11307
McGowan, Jeffrey; Perry, Peter
2002
The divisor of Selberg’s zeta function for Kleinian groups. Appendix A by Charles Epstein. Zbl 1012.11083
Patterson, S. J.; Perry, Peter A.
2001
Asymptotics of the length spectrum for hyperbolic manifolds of infinite volume. Zbl 0986.11059
Perry, P. A.
2001
Isophasal scattering manifolds in two dimensions. Zbl 1005.58011
Brooks, Robert; Perry, Peter A.
2001
On the dimension of the attractor for the non-homogeneous Navier-Stokes equations in non-smooth domains. Zbl 0969.35105
Brown, Russell M.; Perry, Peter A.; Shen, Zhongwei
2000
Isoscattering Schottky manifolds. Zbl 0966.58018
Brooks, Robert; Gornet, Ruth; Perry, Peter
2000
The additive turbulent decomposition for the two-dimensional incompressible Navier-Stokes equations: Convergence theorems and error estimates. Zbl 0922.35123
Brown, Russell M.; Perry, Peter; Shen, Zhongwei
1999
Meromorphic continuation of the resolvent for Kleinian groups. Zbl 0942.58033
Perry, Peter
1999
Isospectral sets for fourth-order ordinary differential operators. Zbl 0912.34068
Caudill, Lester F. jun.; Perry, Peter A.; Schueller, Albert W.
1998
The spectrum of relativistic one-electron atoms according to Bethe and Salpeter. Zbl 0857.47041
Evans, William Desmond; Perry, Peter; Siedentop, Heinz
1996
A trace-class rigidity theorem for Kleinian groups. Zbl 0861.57017
Perry, Peter A.
1995
The Selberg zeta function and scattering poles for Kleinian groups. Zbl 0824.11037
Perry, Peter A.
1995
On Cheeger’s inequality. Zbl 0809.53049
Brooks, Robert; Perry, Peter; Petersen, Peter V
1993
Compactness and finiteness theorems for isospectral manifolds. Zbl 0737.53038
Brooks, Robert; Perry, Peter; Petersen, Peter V
1992
A Mourre estimate and related bounds for hyperbolic manifolds with cusps of non-maximal rank. Zbl 0729.58051
Froese, R.; Hislop, P.; Perry, P.
1991
The Laplace operator on hyperbolic three manifolds with cusps of non- maximal rank. Zbl 0763.58028
Froese, R.; Hislop, P.; Perry, P.
1991
The Selberg zeta function and scattering poles for Kleinian groups. Zbl 0723.11028
Perry, Peter A.
1991
The Selberg zeta function and a local trace formula for Kleinian groups. Zbl 0697.10027
Perry, Peter A.
1990
The Laplace operator on a hyperbolic manifold. II: Eisenstein series and the scattering matrix. Zbl 0677.58044
Perry, Peter A.
1989
Isospectral sets of conformally equivalent metrics. Zbl 0667.53037
Brooks, Robert; Perry, Peter; Yang, Paul
1989
The Laplace operator on a hyperbolic manifold. I: Spectral and scattering theory. Zbl 0631.58030
Perry, Peter A.
1987
Asymptotic completeness of certain four-body Schrödinger operators. Zbl 0589.35032
Hagedorn, George A.; Perry, Peter A.
1986
Commutator methods and Besov space estimates for Schrödinger operators. Zbl 0574.35022
Jensen, Arne; Perry, Peter
1985
Local time decay of high energy scattering states for the Schrödinger equation. Zbl 0538.35026
Cycon, Hans L.; Perry, Peter A.
1985
Multiple commutator estimates and resolvent smoothness in quantum scattering theory. Zbl 0561.47007
Jensen, Arne; Mourre, Éric; Perry, Peter
1984
Exponential bounds and semi-finiteness of point spectrum for N-body Schrödinger operators. Zbl 0546.35051
Perry, Peter A.
1984
A remark on continuum eigenfunctions of N-body Schrödinger operators. Zbl 0576.35107
Perry, Peter A.
1984
Scattering theory by the Enss method. Ed. by B. Simon. Zbl 0529.35004
Perry, Peter A.
1983
Asymptotic completeness for certain three-body Schrödinger operators. Zbl 0506.35026
Hagedorn, George A.; Perry, Peter A.
1983
Spectral analysis of N-body Schrödinger operators. Zbl 0477.35069
Perry, P.; Sigal, I. M.; Simon, B.
1981
Propagation of states in dilation analytic potentials and asymptotic completeness. Zbl 0471.47007
Perry, Peter A.
1981
Mellin transforms and scattering theory I. Short range potentials. Zbl 0445.47009
Perry, Peter A.
1980
all top 5

### Cited by 576 Authors

 28 Perry, Peter A. 15 Sigal, Israel Michael 15 Skibsted, Erik 14 Herbst, Ira W. 11 Tamura, Hideo 10 Guillarmou, Colin 10 Wang, Xueping 9 Borthwick, David 9 Simon, Barry 9 Yafaev, Dimitri R. 8 Ehrenborg, Richard 8 Froese, Richard 8 Gérard, Christian 8 Jensen, Arne 8 Siedentop, Heinz Karl Heinrich 8 Vasy, András 7 Fan, Engui 7 Hryniv, Rostyslav O. 7 Naud, Frédéric 7 Ozawa, Tohru 7 Sá Barreto, Antônio 7 Siltanen, Samuli 7 Soffer, Avraham 6 Adachi, Tadayoshi 6 Albin, Pierre 6 Bytsenko, Andrei A. 6 Gordon, Carolyn S. 6 Liu, Jiaqi 6 Sulem, Catherine 5 Bouclet, Jean-Marc 5 Boutet de Monvel, Anne Marie 5 Brooks, Robert Wolfe 5 Dereziński, Jan 5 Golénia, Sylvain 5 Hislop, Peter D. 5 Isozaki, Hiroshi 5 Klein, Christian 5 Korotyaev, Evgeny L. 5 Liu, Nan 5 Melgaard, Michael 5 Møller, Jacob Schach 5 Topalov, Peter J. 5 Zhong, Chengkui 5 Zworski, Maciej 4 Christiansen, Tanya J. 4 Georgescu, Vladimir 4 Guo, Boling 4 Hassell, Andrew 4 Lassas, Matti J. 4 Matte, Oliver 4 McLaughlin, Kenneth D. T.-R. 4 Muthuramalingam, Pl. 4 Mykytyuk, Yaroslav V. 4 Pohl, Anke D. 4 Rochon, Frédéric 4 Royer, Julien 4 Webb, David L. 4 Weich, Tobias 3 Agmon, Shmuel 3 Bach, Volker 3 Ben-Artzi, Matania 3 Bony, Jean-François 3 Branson, Thomas Patrick 3 Budylin, A. M. 3 Cheng, Qiaoyuan 3 Dyatlov, Semyon 3 Faupin, Jérémy 3 Fröhlich, Jürg Martin 3 Gilkey, Peter B. 3 Guillopé, Laurent 3 Häfner, Dietrich 3 Ito, Hiroshi T. 3 Ito, Kenichi 3 Jakobson, Dmitry 3 Kappeler, Thomas 3 Kazeykina, Anna 3 Koch, Herbert 3 Kovařík, Hynek 3 Łaba, Izabella 3 Lanphier, Dominic 3 Levin, S. B. 3 Lieb, Elliott H. 3 Makover, Eran 3 Manda, Dragos 3 Mandich, Marc-Adrien 3 Marugame, Taiji 3 Matsumoto, Yoshihiko 3 Mazzeo, Rafe R. 3 McOwen, Robert C. 3 Mueller, Jennifer L. 3 Nakamura, Shu 3 Nicoleau, François 3 Robert, Didier 3 Rybkin, Alexei 3 Schlenker, Jean-Marc 3 Sinha, Kalyan B. 3 Soares, Louis 3 Solovej, Jan Philip 3 Stockmeyer, Edgardo 3 Xu, Xingwang ...and 476 more Authors
all top 5

### Cited in 147 Serials

 49 Journal of Functional Analysis 44 Communications in Mathematical Physics 25 Duke Mathematical Journal 23 Journal of Mathematical Physics 23 Communications in Partial Differential Equations 21 Journal of Differential Equations 15 Letters in Mathematical Physics 15 Annales de l’Institut Henri Poincaré. Physique Théorique 14 Journal of Mathematical Analysis and Applications 14 Advances in Mathematics 11 Annales Henri Poincaré 10 Publications of the Research Institute for Mathematical Sciences, Kyoto University 10 The Journal of Geometric Analysis 9 Journal d’Analyse Mathématique 9 Annales de l’Institut Fourier 9 Transactions of the American Mathematical Society 8 Reviews in Mathematical Physics 8 Bulletin of the American Mathematical Society. New Series 7 Inventiones Mathematicae 7 Proceedings of the American Mathematical Society 6 Annales Scientifiques de l’École Normale Supérieure. Quatrième Série 6 Mathematische Zeitschrift 6 Annales de l’Institut Henri Poincaré. Analyse Non Linéaire 6 Geometric and Functional Analysis. GAFA 5 Integral Equations and Operator Theory 5 Differential Geometry and its Applications 5 Discrete and Continuous Dynamical Systems 5 Journal of Spectral Theory 4 Archive for Rational Mechanics and Analysis 4 Inverse Problems 4 Nonlinearity 4 Nuclear Physics. B 4 Mathematische Annalen 4 Nonlinear Analysis. Theory, Methods & Applications. Series A: Theory and Methods 4 Advances in Applied Mathematics 4 Annals of Global Analysis and Geometry 4 Journal of Mathematical Sciences (New York) 4 Communications on Pure and Applied Analysis 3 Discrete Mathematics 3 Acta Mathematica 3 Journal für die Reine und Angewandte Mathematik 3 Nagoya Mathematical Journal 3 Ergodic Theory and Dynamical Systems 3 Physica D 3 Journal de Mathématiques Pures et Appliquées. Neuvième Série 3 Journal of Nonlinear Science 3 Selecta Mathematica. New Series 3 Journal of High Energy Physics 3 Inverse Problems and Imaging 3 Bulletin of Mathematical Sciences 2 Journal of Statistical Physics 2 Mathematical Methods in the Applied Sciences 2 Russian Mathematical Surveys 2 Theoretical and Mathematical Physics 2 Journal of Combinatorial Theory. Series A 2 Journal of Computational and Applied Mathematics 2 Journal of the Mathematical Society of Japan 2 Journal of Soviet Mathematics 2 Proceedings of the Royal Society of Edinburgh. Section A. Mathematics 2 Proceedings of the Indian Academy of Sciences. Mathematical Sciences 2 Experimental Mathematics 2 Mathematical Communications 2 Nonlinear Dynamics 2 Mathematical Physics, Analysis and Geometry 2 Physical Review Letters 2 Journal of Nonlinear Mathematical Physics 2 Comptes Rendus. Mathématique. Académie des Sciences, Paris 2 Central European Journal of Mathematics 2 Journal of the Institute of Mathematics of Jussieu 2 Boundary Value Problems 2 Advances in Mathematical Physics 1 Applicable Analysis 1 Computer Methods in Applied Mechanics and Engineering 1 Communications on Pure and Applied Mathematics 1 Israel Journal of Mathematics 1 Journal of Computational Physics 1 Mathematical Proceedings of the Cambridge Philosophical Society 1 Reports on Mathematical Physics 1 Wave Motion 1 Journal of Geometry and Physics 1 Annales de l’Institut Henri Poincaré. Nouvelle Série. Section A. Physique Théorique 1 Applied Mathematics and Optimization 1 Bulletin of the London Mathematical Society 1 Geometriae Dedicata 1 Illinois Journal of Mathematics 1 Publications Mathématiques 1 Journal of Approximation Theory 1 Manuscripta Mathematica 1 Memoirs of the American Mathematical Society 1 Numerische Mathematik 1 Pacific Journal of Mathematics 1 Proceedings of the Japan Academy. Series A 1 Quarterly of Applied Mathematics 1 Real Analysis Exchange 1 Results in Mathematics 1 Siberian Mathematical Journal 1 Tohoku Mathematical Journal. Second Series 1 Annales de la Faculté des Sciences de Toulouse. Série V. Mathématiques 1 European Journal of Combinatorics 1 Algorithmica ...and 47 more Serials
all top 5

### Cited in 45 Fields

 334 Partial differential equations (35-XX) 196 Quantum theory (81-XX) 151 Operator theory (47-XX) 133 Global analysis, analysis on manifolds (58-XX) 66 Dynamical systems and ergodic theory (37-XX) 50 Differential geometry (53-XX) 31 Ordinary differential equations (34-XX) 25 Number theory (11-XX) 20 Fluid mechanics (76-XX) 16 Combinatorics (05-XX) 16 Functions of a complex variable (30-XX) 16 Relativity and gravitational theory (83-XX) 15 Several complex variables and analytic spaces (32-XX) 14 Numerical analysis (65-XX) 12 Functional analysis (46-XX) 12 Mechanics of particles and systems (70-XX) 12 Optics, electromagnetic theory (78-XX) 8 Statistical mechanics, structure of matter (82-XX) 7 Topological groups, Lie groups (22-XX) 7 Manifolds and cell complexes (57-XX) 6 Algebraic geometry (14-XX) 6 Measure and integration (28-XX) 5 Harmonic analysis on Euclidean spaces (42-XX) 5 Calculus of variations and optimal control; optimization (49-XX) 5 Probability theory and stochastic processes (60-XX) 4 Approximations and expansions (41-XX) 4 Geometry (51-XX) 4 Mechanics of deformable solids (74-XX) 4 Biology and other natural sciences (92-XX) 3 History and biography (01-XX) 3 Nonassociative rings and algebras (17-XX) 3 Difference and functional equations (39-XX) 3 Integral equations (45-XX) 2 Real functions (26-XX) 2 Special functions (33-XX) 2 Abstract harmonic analysis (43-XX) 2 Convex and discrete geometry (52-XX) 2 Information and communication theory, circuits (94-XX) 1 Order, lattices, ordered algebraic structures (06-XX) 1 Linear and multilinear algebra; matrix theory (15-XX) 1 Associative rings and algebras (16-XX) 1 Group theory and generalizations (20-XX) 1 Potential theory (31-XX) 1 Computer science (68-XX) 1 Geophysics (86-XX)