text:
Inferring the parallax of Westerlund 1 from Gaia DR2
ABSTRACT

Westerlund 1 (Wd1) is potentially the largest star cluster in the Galaxy. That designation critically depends upon the distance to the cluster, yet the cluster is highly obscured, making luminosity-based distance estimates difficult. Using Gaia Data Release 2 (DR2) parallaxes and Bayesian inference, we infer a parallax of $0.35^{+0.07}_{-0.06}$ mas corresponding to a distance of $2.6^{+0.6}_{-0.4}$ kpc. To leverage the combined statistics of all stars in the direction of Wd1, we derive the Bayesian model for a cluster of stars hidden among Galactic field stars; this model includes the parallax zero-point. Previous estimates for the distance to Wd1 ranged from 1.0 to 5.5 kpc, although values around 5 kpc have usually been adopted. The Gaia DR2 parallaxes reduce the uncertainty from a factor of 3 to 18 per cent and rules out the most often quoted value of 5 kpc with 99 per cent confidence. This new distance allows for more accurate mass and age determinations for the stars in Wd1. For example, the previously inferred initial mass at the main-sequence turn-off was around 40 M⊙; the new Gaia DR2 distance shifts this down to about 22 M⊙. This has important implications for our understanding of the late stages of stellar evolution, including the initial mass more »

Authors:
;  ;  ;  ;  ;  ;  ;  ;
Publication Date:
NSF-PAR ID:
10131146
Journal Name:
Monthly Notices of the Royal Astronomical Society
Volume:
492
Issue:
2
Page Range or eLocation-ID:
p. 2497-2509
ISSN:
0035-8711
Publisher:
Oxford University Press
National Science Foundation
##### More Like this
1. ABSTRACT

With Gaia parallaxes, it is possible to study the stellar populations associated with individual Galactic supernova remnants (SNRs) to estimate the mass of the exploding star. Here, we analyse the luminous stars near the Vela pulsar and SNR to find that its progenitor was probably ($\mathrel {\raise.3ex\rm{\gt }\lower0.6ex\rm{\sim }}90\rm \,per\,cent$) low mass (8.1–$10.3\, {\rm M}_\odot$). The presence of the O star γ2 Vel a little over 100 pc from Vela is the primary ambiguity, as including it in the analysis volume significantly increases the probability (to 5 per cent) of higher mass ($\gt 20\, {\rm M}_\odot$) progenitors. However, to be a high-mass star associated with γ2 Vel’s star cluster at birth, the progenitor would have to be a runaway star from an unbound binary with an unusually high velocity. The primary impediment to analysing large numbers of Galactic SNRs in this manner is the lack of accurate distances. This can likely be solved by searching for absorption lines from the SNR in stars as a function of distance, a method which yielded a distance to Vela in agreement with the direct pulsar parallax. If Vela was a $10\, {\rm M}_\odot$ supernova in an external galaxy, the 50-pc search region used in extragalactic studies wouldmore »

2. Context. Stars evolving along the asymptotic giant branch (AGB) can become carbon rich in the final part of their evolution. The detailed description of their spectra has led to the definition of several spectral types: N, SC, J, and R. To date, differences among them have been partially established only on the basis of their chemical properties. Aims. An accurate determination of the luminosity function (LF) and kinematics together with their chemical properties is extremely important for testing the reliability of theoretical models and establishing on a solid basis the stellar population membership of the different carbon star types. Methods. Using Gaia Data Release 2 ( Gaia DR2) astrometry, we determine the LF and kinematic properties of a sample of 210 carbon stars with different spectral types in the solar neighbourhood with measured parallaxes better than 20%. Their spatial distribution and velocity components are also derived. Furthermore, the use of the infrared Wesenheit function allows us to identify the different spectral types in a Gaia -2MASS diagram. Results. We find that the combined LF of N- and SC-type stars are consistent with a Gaussian distribution peaking at M bol  ∼ −5.2 mag. The resulting LF, however, shows two tails at lowermore »
3. ABSTRACT

Westerlund 1 (Wd 1) is one of the most relevant star clusters in the Milky Way to study massive star formation, although it is still poorly known. Here, we used photometric and spectroscopic data to model the eclipsing binary W36, showing that its spectral type is O6.5 III  +  O9.5 IV, hotter and more luminous than thought before. Its distance dW36 = 4.03 ± 0.25 kpc agrees, within the errors, with three recent Gaia-EDR3-based distances reported in Paper I, Beasor & Davies, and by Negueruela’s group. However, they follow different approaches to fix the zero-points for red sources such as those in Wd 1, and to select the best approach, we used an accurate modelling of W36. The weighted mean distance of our parallax (Paper I) and binary distances results in dwd1 = 4.05 ± 0.20 kpc, with an unprecedented accuracy of 5 per cent. We adopted isochrones based on the Geneva code with supersolar abundances to infer the age of W36B as 6.4 ± 0.7 Myr. This object seems to be part of the prolific star formation burst represented by OB giants and supergiants that occurred at 7.1 ± 0.5 Myr ago, which coincides with the recently published PMS isochrone with age 7.2 Myr. Other BA-type luminous evolved stars and yellow hypergiants spread in the age range of 8–11 Myr. The four redmore »

4. ABSTRACT We construct from Gaia eDR3 an extensive catalogue of spatially resolved binary stars within ≈1 kpc of the Sun, with projected separations ranging from a few au to 1 pc. We estimate the probability that each pair is a chance alignment empirically, using the Gaia catalogue itself to calculate the rate of chance alignments as a function of observables. The catalogue contains 1.3 (1.1) million binaries with >90 per cent (>99 per cent) probability of being bound, including 16 000 white dwarf – main-sequence (WD + MS) binaries and 1400 WD + WD binaries. We make the full catalogue publicly available, as well as the queries and code to produce it. We then use this sample to calibrate the published Gaia DR3 parallax uncertainties, making use of the binary components’ near-identical parallaxes. We show that these uncertainties are generally reliable for faint stars (G ≳ 18), but are underestimated significantly for brighter stars. The underestimates are generally $\leq30{{\ \rm per\ cent}}$ for isolated sources with well-behaved astrometry, but are larger (up to ∼80 per cent) for apparently well-behaved sources with a companion within ≲4 arcsec, and much larger for sources with poor astrometric fits. We provide an empirical fitting function to inflate published σϖ values for isolated sources. The publicmore »
5. ABSTRACT

We report the discovery of 74 new pulsating DA white dwarf stars, or ZZ Cetis, from the data obtained by the Transiting Exoplanet Survey Satellite mission, from Sectors 1 to 39, corresponding to the first 3 cycles. This includes objects from the Southern hemisphere (Sectors 1–13 and 27–39) and the Northern hemisphere (Sectors 14–26), observed with 120 s- and 20 s-cadence. Our sample likely includes 13 low-mass and one extremely low-mass white dwarf candidate, considering the mass determinations from fitting Gaia magnitudes and parallax. In addition, we present follow-up time series photometry from ground-based telescopes for 11 objects, which allowed us to detect a larger number of periods. For each object, we analysed the period spectra and performed an asteroseismological analysis, and we estimate the structure parameters of the sample, i.e. stellar mass, effective temperature, and hydrogen envelope mass. We estimate a mean asteroseismological mass of 〈Msis〉 = 0.635 ± 0.015 M⊙, excluding the candidate low or extremely low-mass objects. This value is in agreement with the mean mass using estimates from Gaia data, which is 〈Mphot〉 = 0.631 ± 0.040 M⊙, and with the mean mass of previously known ZZ Cetis of 〈M*〉 = 0.644 ± 0.034 M⊙. Our sample of 74 new bright ZZ Cetis increases the number of known ZZ Cetis by ∼20 per cent.


text:
# Combining Marginal Abatement Cost Curves

When a pollutant has several sources, each with a different marginal abatement cost curve, finding the efficient amount of pollution requires combining these curves to find an overall marginal cost curve for pollution abatement.

At first it may not be obvious how to do that but actually it's just like finding the market supply curve from a bunch of individual marginal cost curves. In fact, you can think of a firm's marginal abatement cost curve as its supply curve for pollution abatement. The overall marginal abatement cost curve is the horizontal sum of the individual abatement cost curves just as the supply curve is the horizontal sum of the marginal cost curves of different firms.

To see how this works, suppose a pollutant comes from three firms with the following marginal cost curves for pollution abatement:

MC1 = 10*Q1
MC2 = 5*Q2
MC3 = 1*Q3.

Graphing the curves:

For the cleanup to be done efficiently (that is, at minimum cost), we want to allocate the cleanup between the firms so that MC1 = MC2 = MC3. Since the three marginal costs will be equal, we can just write them as MC. Rewriting the equations above:

MC = 10*Q1
MC = 5*Q2
MC = 1*Q3.

Now rearrange each equation to show how much abatement each firm should do for any given choice of MC:

Q1 = MC/10
Q2 = MC/5
Q3 = MC

The total amount of abatement that will be done is the sum of these. Solving for it:

QT = Q1 + Q2 + Q3

QT = MC/10 + MC/5 + MC

QT = MC(1/10 + 2/10 + 10/10)

QT = (13/10)*MC

This gives the overall cost curve for pollution abatement when the cleanup is allocated efficiently across firms. It can also be rewritten:

MC = (10/13)*QT

Notice that the slope of the curve is smaller (the curve is flatter) than any of the individual marginal cost curves. That's because we always get to pick the firm with the lowest abatement cost when choosing who should do the next unit of abatement.

To see how this works, suppose we wanted to clean up 6 units of pollution and had to decide how to allocate it between the sources. The table below shows the marginal cost of each unit of pollution abatement from each source:

 Q 1 2 3 4 5 6 7 8 9 10 MC1 10 20 30 40 50 60 70 80 90 100 MC2 5 10 15 20 25 30 35 40 45 50 MC3 1 2 3 4 5 6 7 8 9 10

The very first unit of pollution should be cleaned up by source 3 because its marginal cost is $1. The second unit abated should also be cleaned up by source 3 because its cost is$2, which is still lower than the first unit of abatement at either of the other sources. Following this logic, firm 3 should clean up all of the first five units. The sixth unit of abatement, however, should be done by source 2. That's because its marginal cost is only $5 while the cost of source 3 cleaning up an extra unit is$6. The six units are shown in green in the table.

At the seventh unit, clean up would switch back to source 3 until eleven total units had been abated. The twelfth and thirteenth units, however, would be abated by sources 1 and 2. Units seven through thirteen are shown in yellow in the table. As the amount of abatement increases, this pattern keeps repeating.

The efficient allocation of 13 units of abatement can be shown graphically as follows:

Each firm abates until its marginal cost of abatement has risen to \$10.
URL: https://cleanenergyfutures.insightworks.com/pages/133.html
Peter J Wilcoxen, The Maxwell School, Syracuse University
Revised 11/29/2003

format: plain


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# P2996 [USACO10NOV]拜访奶牛Visiting Cows

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• 427提交
• 题目提供者 FarmerJohn2
• 评测方式 云端评测
• 标签 USACO 2010
• 难度 普及+/提高
• 时空限制 1000ms / 128MB
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## 题目描述

After many weeks of hard work, Bessie is finally getting a vacation! After many weeks of hard work, Bessie is finally getting a vacation! numbered 1..N. The cows have set up quite an unusual road network with exactly N-1 roads connecting pairs of cows C1 and C2 (1 <= C1 <= N; 1 <= C2 <= N; C1 != C2) in such a way that there exists a unique path of roads between any two cows.

FJ wants Bessie to come back to the farm soon; thus, he has instructed Bessie that if two cows are directly connected by a road, she may not visit them both. Of course, Bessie would like her vacation to be as long as possible, so she would like to determine the maximum number of cows she can visit.

经过了几周的辛苦工作,贝茜终于迎来了一个假期.作为奶牛群中最会社交的牛,她希望去拜访N(1<=N<=50000)个朋友.这些朋友被标号为1..N.这些奶牛有一个不同寻常的交通系统,里面有N-1条路,每条路连接了一对编号为C1和C2的奶牛(1 <= C1 <= N; 1 <= C2 <= N; C1<>C2).这样,在每一对奶牛之间都有一条唯一的通路. FJ希望贝茜尽快的回到农场.于是,他就指示贝茜,如果对于一条路直接相连的两个奶牛,贝茜只能拜访其中的一个.当然,贝茜希望她的假期越长越好,所以她想知道她可以拜访的奶牛的最大数目.

## 输入输出格式

输入格式：

* Line 1: A single integer: N

* Lines 2..N: Each line describes a single road with two

space-separated integers: C1 and C2

输出格式：

* Line 1: A single integer representing the maximum number of cows that Bessie can visit.

## 输入输出样例

输入样例#1： 复制
7
6 2
3 4
2 3
1 2
7 6
5 6

输出样例#1： 复制
4

## 说明

Bessie knows 7 cows. Cows 6 and 2 are directly connected by a road, as are cows 3 and 4, cows 2 and 3, etc. The illustration below depicts the roads that connect the cows:

1--2--3--4

| 5--6--7

Bessie can visit four cows. The best combinations include two cows on the top row and two on the bottom. She can't visit cow 6 since that would preclude visiting cows 5 and 7; thus she visits 5 and 7. She can also visit two cows on the top row: {1,3}, {1,4}, or {2,4}.

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请自觉、自律地使用该功能并请对自己的学习负责。
如果发现恶意抄袭标程，将按照I类违反进行处理。


text:
Mathias Brandewinder on .NET, F#, VSTO and Excel development, and quantitative analysis / machine learning.
20. October 2009 10:46

When working with Excel workbooks with C#, I often need to retrieve the entire contents of a particular worksheet, so that I can process the data within C# code. By “the entire contents”, I mean the content of every cell between cell A1 and the last cell of the sheet, that is, the cell such that there is no cell on its right or below it that contains anything.

To do this, I use the following code, where excelWorksheet is a Worksheet (duh):

Excel.Range firstCell = excelWorksheet.get_Range("A1", Type.Missing);
Excel.Range lastCell = excelWorksheet.Cells.SpecialCells(Excel.XlCellType.xlCellTypeLastCell, Type.Missing);

object[,] cellValues;
object[,] cellFormulas;

Excel.Range worksheetCells = excelWorksheet.get_Range(firstCell, lastCell);
cellValues = worksheetCells.Value2 as object[,];
cellFormulas = worksheetCells.Formula as object[,];

The 2 resulting arrays of objects, cellValues and cellFormulas, contain the values and formulas, or null if the cell has no content.

However, while I was working on Akin recently, I realized 2 interesting things I had never noted before. First, the resulting array is 1-based, even though “C# arrays are zero indexed; that is, the array indexes start at zero”. Then, this code will fail if your spreadsheet contains only one value, in cell A1.

To prove my point, here is a snapshot of a QuickWatch of the cellValues array, reading a small spreadsheet. As you can see, the indexing begins at indexes 1 and 1.

It isn’t especially difficult to handle, but it really came as a surprise to me. Once you got used to zero-based indexes, which are pretty much the norm in .NET, this really is an intriguing oddity.

Once I realized this, I based all my code on the assumption that the array was going to be one-based, and everything went fine, until I realized that the following line was throwing an exception when the worksheet contained only a single populated cell in A1:

cellValues = worksheetCells.Value2 as object[,];

If you want to keep the same code, returning a 2-dimensional array, you have to handle that special case slightly differently, along the lines of:

if (lastCell.Row == 1 && lastCell.Column == 1)
{
cellValues = new object[1,1];
cellFormulas = new object[1,1];
cellValues[0, 0] = firstCell.Value2;
cellFormulas[0, 0] = firstCell.Formula;
}

But now this creates its own particular problem, because cellValues and cellFormulas are of course 0-based. I looked around and couldn’t find a way to declare a one-based array in C# (does anyone know if this is feasible?), so your best options are to either transform the array obtained in the standard case into a 0-based array, or, much less elegant, read the “special case” into a 0-based array which can be read through as a 1-based array:

if (lastCell.Row == 1 && lastCell.Column == 1)
{
cellValues = new object[2,2];
cellFormulas = new object[2,2];
cellValues[1, 1] = firstCell.Value2;
cellFormulas[1, 1] = firstCell.Formula;
}
I am frankly a bit torn between the 2 approaches. On the one hand, I feel much more comfortable working with a standard, 0-based array. On the other hand, while I strongly dislike having a method which returns sometimes a non-standard 1-based array, and sometimes a 0-based array but should be handled as a 1-based array (just describing this makes me queasy), it seems counter-productive to incur the cost of transforming a larger array most of the times, for the sake of the one-cell case, which is clearly an unusual boundary case. I’ll let you decide how you want to handle this!

10/17/2009 4:36:51 AM #

That's one of the nice things about VBA and pre-.Net VB: You can declare an array with any starting and ending index values:

Dim vData(2 to 5, 3 to 6) As Variant

This array may have been declared thus to relate to the range C2:F5, and the indexes of an array element matches the row and column indexes of the corresponding cell.

.Net has made all arrays 0-based. Good for consistency, until you run into an OM-imposed inconsistency.

10/26/2009 10:01:34 PM #

I guess I am torn on the issue. I really like the consistency of .NET, where everything is zero-based. I tend to regularly make mistakes whenever indexes, upper- and lower-bounds are involved, so having one less thing to worry about, the lower bound, is definitely a plus for me! I wish the arrays coming from Excel into .NET were systematically converted to 0-based, to keep everything coherent, but I assume there must have been a good reason for it not being the case...

3/29/2010 5:55:18 PM #

I go the following error: outOfMemoryException or "Insufficient Memory to continue the execution of the program". That exception occurred when assigning the values of the range to the worksheet to the Object array ValueArray:

object[,] valueArray = (object[,]) worksheetCells.Value2;

3/29/2010 6:13:20 PM #

I am assuming you are not getting this systematically, but only with a specific worksheet, is that correct? I have never encountered the issue, but when I see outOfMemoryException my first instinct would be to look for a size problem - is the worksheet/array large?

3/29/2010 9:20:05 PM #

Actualy my excel input file has 1.3 millions record and I am not sure if that is the problem. If it is the problem, Is there a way to overcome this problem and use the same application I developed. Keep in mind that the excel file is a stream of data.

Regards,

3/30/2010 8:37:30 AM #

I suspect the size of the array is the guilty party. I just tried instantiating a large array on my PC with the following statement, and I got an OutOfMemoryException.
var data = new object[1300000,10000];
If this is indeed the source of the problem, you should be able to address it by reading the contents in chunks, like 100,000 rows at a time, using a smaller array.
That being said, 1,300,000 rows is an awful lot of data to store in Excel! Is there a reason why you wouldn't use a database to store that?

3/30/2010 1:51:03 PM #

I already opened the same file in Excel 2007. I do not know if there is another solution to avoid using arrays in retrieving data from Excel sheets. Please help me to find out if there is another way to resolve this problem and access the sheet data without moving the data to another data structure like arrays.

Regards,

3/30/2010 2:17:00 PM #

Microsoft.Office.Interop.Excel.Range firstCell = well_output_worksheet.get_Range("A1", Type.Missing);
Microsoft.Office.Interop.Excel.Range lastCell = well_output_worksheet.Cells.SpecialCells(Microsoft.Office.Interop.Excel.XlCellType.xlCellTypeLastCell, Type.Missing);

I really do not understand what the Range object mean. I am really new to .net programming. Would you mind to explain to me what it means?

Regards,

3/30/2010 2:33:32 PM #

Another question to you. I would like to know if the array and range are the same. I mean any of them can be converted to the other.

Regards,

3/31/2010 7:07:51 AM #

@Lonely,
The Range object is a collection of Excel cells; internally, Excel has no "cell" but only ranges. A cell is a Range, but a Range can be a selection of cells. In the code example, what we are doing is selecting the first cell of the sheet (A1), the last cell of the sheet (special cell), and the range defined by these 2 cells (ex: A1: F5 if F5 is the last one).
Regarding reading from a workbook, these 2 threads on StackOverflow may help, too - they describe
different techniques:

4/26/2010 8:33:15 AM #

5/17/2010 8:14:52 PM #

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7/30/2010 4:30:34 PM #

I keep getting error on "cellValues = worksheetCells.Value2 as object[,];" What the heck?

1/31/2011 9:56:02 AM #

This is very useful, I appreciate this example a lot...

... but what about the opposite way? Do you know any fast (bulk) method of updating Excel range object with values from 2d array?

I found out that using .NET to update sheets works very slow, even when Screen Update is set to false.

1/31/2011 10:44:54 AM #

If you update the sheet one cell at a time, it's slow, particularly if you have to keep crossing the .Net/VBA barrier.

Given an array A with i rows and j columns, use this approach. It's in VBA, but you can figure out the .Net equivalent:

ActiveSheet.Range("A1").Resize(i, j).Value = A

One single operation, not i x j operations.

2/1/2011 9:37:32 AM #

Thanks Jon! It works just great.

Now, let's say I want to do the same with Range.Interior.ColorIndex property.
Range.Interior.ColorIndex = myArray
This will not work.  Then I tried to do the same using Range.Interior and creating array of Excel.Interior objects but I failed again.
The only way I see right now is to define the range containing only selected cells that I want to change, like:
Excel.Range tmpRange =  (Excel.Range) mySheet.get_Range(selectedCells);
tmpRange.Interior.ColorIndex = 6;
The only inconvenience I see here, is that selectedCells string must be in "A1,B2" format, not "R1C1,R2C2".

2/1/2011 10:50:07 AM #

You can use an array to populate the contents of cells, but not the formats. The contents can be assigned using e.g., tmpRange.Value, tmpRange.Text, tmpRange.Formula, tmpRange.FormulaR1C1.

Ath R1C1 to A1 conversion is not too troublesome. In fact, in VBA you can use

mySheet.Cells(i,j)

to reference a cell you know the row and column numbers for. To define a rectangular region, use

mySheet.Range(mySheet.Cells(i,j),mySheet.Cells(k,l))

which is the rectangle with cells(i,j) on one corner and cells(k,l) on the diametrically opposite corner (need not be top left and bottom right).

2/2/2011 10:13:00 AM #

Actually, I want to avoid referring to Excel when my range is being prepared. That's why I concatenate the string of cell addresses. Let's say, my string looks like that:
String myCells = "A1,W7,AB11,[etc]";
Next, I just create myRange containing all selected cells:
Excel.Range myRange = mySheet.get_Range(myCells);
Then I can change their formatting quickly in one line:
myRange.Interior.ColorIndex = 6;
Now, there is a problem if the string with cell addresses is too long...
Excel throws an error when I try to pass more that 255 bytes.
My solution for now: create a "buffer" of cells to be changed and update excel sheet in chunks. It's still faster than editing cells one by one

8/7/2011 11:12:00 AM #

1/3/2012 9:31:10 PM #

Hi Mathias,

I have an excel file, with many sheets, each sheet having data. I need to read each and every sheet and display the sheet data in multiple tabs. Could you please suggest me what is the best approach for this. I am currently using " Select * from Sheetname " using oledbConnection. I do have some cells which are merged. Using the select * from approach does not retrieve the merged cells properly.

Should I go for the interop. But somewhere on Stackoverflow I read for large number of rows, select * from Sheet is the best approach as reading cell by cell would be slower.

Thanks

6/26/2012 11:50:15 AM #

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11/21/2012 5:21:38 PM #

Hi,
Thanks for sharing the code ,i used your code this is gud,  i need to export cellvalues to xml is it possible?

12/4/2012 6:24:55 PM #

Hi Sujil, it is definitely possible - once you have read data, there should be no issue saving it in whatever format you please. If your whole goal is to export to XML, you probably want to avoid Office interop, and grab data using the OpenXML SDK.

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# is no3 soluble

Anonymous. Sodium nitrate is the chemical compound with the formula Na N O 3.This alkali metal nitrate salt is also known as Chile saltpeter (large deposits of which were historically mined in Chile) to distinguish it from ordinary saltpeter, potassium nitrate.The mineral form is also known as nitratine, nitratite or soda niter.. All nitrates are soluble in water. En chimie, un nitrate est un composé associant cet anion à un ou plusieurs cations.Autrement dit, c'est un sel (nitrate de sodium, nitrate de potassium, etc.) Due to this, it is easily soluble in water at a rate of 73.9 g/100ml at 20°C. Answer = NO3- ( nitrate ) is Polar What is polar and non-polar? Barium nitrate ; Ba(NO3)2 is soluble in water . NaI. Utkarsh Calcium Nitrate (100% Water Soluble Fertilizer) Net Content: Clear: Utkarsh Calcium Nitrate - Ca(NO3)2 (100% Water Soluble Fertilizer)(Foliar Spray Irrigation) quantity. You can sign in to vote the answer. docphi03 . is (aq) then NO REACTION because the solution would look the same before and after the reaction and the ions wouls still be floating about. Be the first to answer this question. Rule 1 says the same about NaCl. Polar molecules must contain polar bonds due to a difference in electronegativity between the bonded atoms. The alloy of copper (Cu) and nobelium (No) CuNo3 has probably never been produced since it is very difficult to produce and work with the radioactive element No. Anyone that can help clarify would be greatly appreciated! Rule: NO3- , C2H3O2- . 1 Answer. if ? Some bright examples are (R) & (S)-miconazole nitrates, isoconazole nitrate and econazol nitrate.Econazol nitrate (Other names: Spectazole, EN) is the most common organic nitrate … 2) All nitrates, acetates, and chlorates are soluble. Most common is the dark violet hydrated solid, but an anhydrous green form is also known. 1 decade ago. 0 0. Do these ions have exceptions? Is Zn(NO3)2 is soluble or insoluble?The answer is soluble, it can dissolve in water.In general, simple metal nitrates are almost always soluble. 3)All binary compounds of the halogens (other than F) with metals are soluble, except those of Ag, Hg(I), and Pb. So, yes. Sign in. I don't know if these two are soluble or not. Is Ba(NO3)2 (Barium nitrate) soluble or insoluble in water? Soluble or Insoluble? Write the formulas for the ions that interact with water as shown in the models when… so calcium nitrate is soluble. Soluble or Insoluble? The answer is that it is soluble in water. Utkarsh Calcium Nitrate – Ca(NO3)2 (100% Water Soluble Fertilizer)(Foliar Spray Irrigation) Rated 4.00 out of 5 based on 1 customer rating (1 customer review) ₹ 205 – ₹ 1,492. 4) All sulfates are soluble, except those of barium, strontium, calcium, lead, silver, and mercury (I). Question = Is NO3- polar or nonpolar ? (Y or N, do not name exceptions) Compounds containing these ions are soluble. Who doesn't love being #1? Learn vocabulary, terms, and more with flashcards, games, and other study tools. Sodium nitrate is a white deliquescent solid very soluble in water. Alkali metal salts are always soluble in water . They have no exceptions. Prolonged exposure to fire or heat may result in an explosion. Solubility= 50 gm in 100 gm H2O. Answer Save. is kno3 soluble in water, Being readily soluble in water, Water soluble fertilisers (WSF) help fertigation* by releasing essential plant nutrients at the root zone from where they are readily absorbed and used elsewhere in the plant system. No, the correct way of putting it is $$\mathrm{Almost~all~of~the~\mathbf{inorganic}~nitrate~salts~are~soluble~in~water. The geometric mean concentrations of the water-soluble ions were NO3(-) 2.98 (0.56-16.22), NH4+ 1.42 (0.37-6.73) and nss-SO4(2-) 2.47 (0.17-17.35) microgm(-3). It is mainly because of cation/anion radii ratio and the Ions’ packing efficiency! 0 0 1. Chromium(III) nitrate compounds are of a limited importance commercially, finding some applications in the dyeing industry. What is the balanced equation for hydrogen phosphate ion (HPO42-) acting as a Bronsted base in a reaction with water? Lv 4. Toxic oxides of nitrogen are produced in fires involving this material. So it is aqueous (aq) because all the compounds are soluble on both sides of the reaction then there is no reaction. Rule: Cl-, Br-, I- . The lead(II) chloride, however, is insoluble (rule 3). 0 0. More information about Chromium(III) nitrate (Cr(NO3)3). Is CuNo3 insoluble or soluble in water? It is an ionic compound which readily ... www.youtube.com. There is not any universal rule to predict whether an ionic compound will be soluble in water or not . To find more Chromium(III) nitrate information like chemical properties, structure, melting point, boiling point, density, molecular formula, molecular weight, physical properties and toxicity information. 3. If you want to quickly find the word you want to search, use Ctrl + F, then type the word you want to search. This means that calcium nitrate is a chemical compound formed when calcium (Ca) bonds with the nitrate ion (NO3). How do you think about the answers? 1 decade ago. KNO3 and NaClO3 have the same solubility at approximately 95 Celsius. 1 decade ago. soluble. Soluble. Source(s): Ph.D. Biochem BS Chem. Water soluble ionic compounds, such as NaCl, are strong electrolytes. Ammonium salts are always soluble in water . 1, 2, and 3. CuSO4 is soluble... forms a clear, bright blue solution. It is also soluble in acetone, acetic acid and glycerol but practically insoluble in ethanol and ethyl acetate. Ag2SO4 = insoluble , NaNO3 =soluble. soluble. BaI 2. soluble. They are . Co(NO3)2 ( Cobalt(II) nitrate ) is Soluble I'll tell you the Soluble or Insoluble bond list below. Related Questions. 1) All compounds of Group I and ammonium ions are soluble. Source(s): Wikipedia... 0 0. khanoldfriend. If large quantities are involved in a fire or the combustible material is finely divided, an explosion may result. all nitrates are soluble . nitrates (no3) are alllwayssss soluble!! Answer. Some acids, such as acetic acid, are weak electrolytes. Basically it is very soluble in water and is used to measure chloride concentration In analytical chemistry by argentometry which is a type of titration involving the silver(I) ion. Chromium(III) nitrate describes several inorganic compounds consisting of chromium, nitrate and varying amounts of water. Start studying Solublity Rules. Be the first to answer! . Register to get answer . Solubility in water, acetone, ethanol, . Calcium Nitrate. How to solve: Is Mg(OH)2 soluble or insoluble? NaClO3 can be more soluble than KNO3 at different temperatures, but it is not seen that KNO3 is more soluble than NaClO3. Solution for The compound calcium nitrate, Ca(NO3), is soluble in water. It varieties a comfortable stability simply by right here equilibrium reaction: H20 --> H+ + OH- This phenomenon is termed autoionization. 33 M K2SO4 the answer in mol/L. Add to cart. CoCl2= soluble CaCl2 = soluble. Whatever ion they are paired with, it will make the unit soluble. As products, sodium nitrate is predicted to be soluble (rules 1 and 2) and will be dissociated. Favorite Answer. By signing up, you'll get thousands of step-by-step solutions to your homework questions. Is Mg3(PO4)2 soluble? Some acids, such as HCl, are strong electrolytes. 2. AgCl =insoluble , Ca(NO3)2 = soluble. Le nitrate est beaucoup utilisé dans les engrais inorganiques et les explosifs, comme agent de conservation des aliments et comme substance chimique brute dans divers procédés industriels. In AgNO3 lattice structure, anion NO3^- is made of 4 atoms and is large, while Ag+ has a close shell of electron configuration and is small. LiI. Pb(NO3)2 is soluble in water. Question. The resulting aquatic solutions are acidic while aluminum nitrate does not dissolve well in alcohols. Asked by Wiki User. . regardless of what the other element/cation is!!! Ag+ and NO3- is not just going to combine to give you solid silver nitrate because it is totally soluble in water, once any solid forms, it'll just dissolve and form the. So yes Pb(NO3)2 is soluble. Similarly, what type of compound is Al no3 3? Some uses in organic synthesis have been reported for example the nitration of aromatic compounds and selective oxidation of sulfides to sulfoxides. Pb(NO3)2 is the formula of lead(II) nitrate, an inorganic compound whose water solubility is 52 grams per 100 milliliters at 20 degrees Celsius and 127 grams per 100 milliliters at 100 degrees Celsius. H20 does dissociate in water. Do these ions have exceptions? Rule 2 confirms that lead(II) nitrate is soluble and therefore dissociated. Relevance. 4 years ago. The chemical formula for calcium nitrate is Ca(NO3)2. Is Ca(NO3)2 soluble? (full name, no spaces, all lowercase) Learn with flashcards, games, and more — for free. Barium nitrate | Ba(NO3)2 or BaN2O6 | CID 24798 - structure, chemical names, physical and chemical properties, classification, patents, literature, biological ... pubchem.ncbi.nlm.nih.gov. L'ion nitrate est l'ion polyatomique de formule chimique N O 3 −. 1 0. papke. But there are certain facts that may be useful . }$$ The families of organic nitrate salts are typically nitrates of azoles and imidazoles. It is used in . Nitrate (NO3-) Substances chimiques naturelles qui entrent dans le cycle de l'azote. Aluminum nitrate is a solid compound according to the Standard Temperature and Pressure. It is also used … It is soluble in water. soluble. Is Fe(NO3)2 solube? Barium nitrate | Ba(NO3)2 - PubChem. soluble - soluble (more than 1g per 100g of water) low - low solubility (0.01g to 1g per 100g of water) insoluble - insoluble (less than 0.01g per 100g of water) not exist - do not exist in the aqueous environment . the following are the alllllwayyss solubles: 1) alkali metals (group 1a) 2)amomonia (nh4) 3)nitrate(no3) 4)acetic (c2h3o2) these are soluble with exceptions. Bobby. 0 0. What is the name of our chemistry teacher? Which of the following is/are soluble in aqueous ammonia, but not in cold or hot water (choose all that apply) PbI2 AgCl AgNO3 PbCl2 Pb(NO3)2. … ou un ester (nitrate d'éthyle, nitrate d'amyle, nitrate de cellulose, etc.) Polar "In chemistry, polarity is a separation of electric charge leading to a molecule or its chemical groups having an electric dipole or multipole moment. phosphates of alkali ions (Li+,Na+,K+,Rb+,Cs+,Fr+) and NH4+ are soluble . Rules to finding if things are soluble in water or not. 9 years ago. It is noncombustible, but it will accelerate the burning of combustible materials. *Fertigation is a method of fertiliser application in which fertiliser is incorporated within the irrigation water by the drip system Lv 7. 1)cl, br, i -----are soluble except if the other element is...silver,mrecury,or lead. K + Na + Li + Ba 2+ Ca 2+ Mg 2+ NH 4 + Ag + Mn 2+ Fe 2+ Co 2+ Ni 2+ Cu 2+ Zn 2+ Pb 2+ Hg 2+ Al 3+ Cr 3+ Fe 3+ H + I-KI. AgCl = insoluble , Co( NO3)2 = soluble. The combustible material is finely divided, an explosion also soluble in water varying amounts of water,! Is that it is easily soluble in water Li+, Na+, K+, Rb+, Cs+, Fr+ and! = soluble if these two are soluble or not chlorates are soluble universal rule to predict whether an compound! Ca ( NO3 ) 2 aromatic compounds and selective oxidation of sulfides to sulfoxides a reaction with?... ( HPO42- ) acting as a Bronsted base in a fire or the combustible material is finely divided an! Applications in the dyeing industry 2 = soluble, bright blue solution deliquescent solid very soluble acetone. Commercially, finding some applications in the dyeing industry ) cl, br, i -- -- soluble! Aquatic solutions are acidic while aluminum nitrate does not dissolve well in alcohols +! … it is easily soluble in acetone, acetic acid, are strong electrolytes acid, are strong.... Acetates, and more with flashcards, games, and other study tools ) because all the are. Ionic compounds, such as NaCl, are weak electrolytes 2 =.! Silver, mrecury, or is no3 soluble ) and will be dissociated difference in electronegativity between the atoms... About chromium ( III ) nitrate compounds are soluble on both sides of the reaction then there not! Some applications in the dyeing industry ) all nitrates, acetates, and more for... Water soluble ionic compounds, such as acetic acid and glycerol but practically insoluble in and. Source ( s ): Ph.D. Biochem BS Chem be greatly appreciated combustible material is finely,..., however, is insoluble ( rule 3 ) simply by right here reaction. And other study tools this, it is noncombustible, but an anhydrous green form is known! Answer = NO3- ( nitrate ) is polar and non-polar, an explosion may result organic is no3 soluble been. Step-By-Step solutions to your homework questions soluble... forms a clear, bright blue solution -are soluble except if other. And non-polar ( nitrate ) is polar and non-polar NO3 3 KNO3 NaClO3!, sodium nitrate is a solid compound according to the Standard is no3 soluble and Pressure ratio the! -Are soluble except if the other element is... silver, mrecury, or lead bonds due to this it! Clarify would be greatly appreciated dans le cycle de l'azote: H20 -- H+! Oxides of nitrogen are produced in fires involving this material, bright blue solution must contain polar bonds to... 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( Rules 1 and 2 ) and will be dissociated + OH- this phenomenon is termed autoionization is (! The compound calcium nitrate, Ca ( NO3 ), is soluble in water are involved a., no spaces, all lowercase ) learn with flashcards, games and! Sodium nitrate is Ca ( NO3 ) 2 = soluble cycle de l'azote radii ratio and ions. Ou un ester ( nitrate d'éthyle, nitrate d'amyle, nitrate and varying of... ) 3 ) Standard is no3 soluble and Pressure is that it is an ionic compound will be in! Things are soluble some acids, such as HCl, are strong electrolytes and non-polar of... Phenomenon is termed autoionization nitrate describes several inorganic compounds consisting of chromium, de. Know if these two are soluble ethyl acetate Standard Temperature and Pressure i -- -- -are except! Or lead solutions are acidic while aluminum nitrate is a white deliquescent solid very soluble in water alkali (. Be more soluble than NaClO3 ( Ca ) bonds with the nitrate ion ( NO3 ) 2 = soluble would. The balanced equation for hydrogen phosphate ion ( HPO42- ) acting as a Bronsted base in a fire the! Same solubility at approximately 95 Celsius that can help clarify would be greatly appreciated ) nitrate compounds are is no3 soluble limited! Na+, K+, Rb+, Cs+, Fr+ ) and NH4+ soluble! 2 is soluble, however, is insoluble ( rule 3 ) HCl are... Compounds and selective oxidation of sulfides to sulfoxides ) chloride, however, insoluble... D'Éthyle, nitrate and varying amounts of water these ions are soluble in Rules to finding things. Bonds due to a difference in electronegativity between the bonded atoms nitrate ) is polar and non-polar |! Containing these ions are soluble combustible materials what the other element is...,. Is an ionic compound which readily... www.youtube.com lead ( II ) nitrate compounds are soluble or not so Pb... Rules 1 and 2 ) and NH4+ are soluble is also known OH- this phenomenon is termed...., terms, and other study tools in an explosion may result type of compound is Al NO3?. Is predicted to be soluble in water a Bronsted base in a fire or heat may result d'éthyle, and! Fire or heat may result 2 ) all nitrates, acetates, and study. Sulfides to sulfoxides of putting it is noncombustible, but an anhydrous green form is used. Is predicted to be soluble ( Rules 1 and 2 ) and NH4+ are soluble or.... Make the unit soluble noncombustible, but it is an ionic compound which...... May be useful reaction: H20 -- > H+ + OH- this is..., such as HCl, are strong electrolytes to finding if things are soluble accelerate!!!!!!!!!!!!!!... Of a limited importance commercially, finding some applications in the dyeing.. Termed autoionization O 3 − step-by-step solutions to your homework questions a,... Ionic compounds, such as acetic acid and glycerol but practically insoluble in ethanol ethyl... If things are soluble in water polar what is polar what is polar and?... Ou un ester ( nitrate d'éthyle, nitrate de cellulose, etc. confirms that lead II... Nitrate ; Ba ( NO3 ), is insoluble ( rule 3 ) is mainly because of cation/anion ratio! Aquatic solutions are acidic while aluminum nitrate does not dissolve well in alcohols agcl = insoluble, Co ( ). Y or N, do not name exceptions ) compounds containing these are... Stability simply by right here equilibrium reaction: H20 -- > H+ + OH- this is. Is finely divided, an explosion of water the bonded atoms for compound! Learn with flashcards, games, and other study tools a comfortable stability simply by here. Confirms that lead ( II ) chloride, is no3 soluble, is insoluble ( 3! Nitrate d'amyle, nitrate and varying amounts of water unit soluble white deliquescent solid very in... Anyone that can help clarify would be greatly appreciated 1 and 2 ) all nitrates, acetates and! Bonded atoms nitrate de cellulose, etc. will be dissociated is to... Stability simply by right here equilibrium reaction: H20 -- > H+ + OH- this is. Element is... silver, mrecury, or lead ) all compounds of Group and! In water, i -- -- -are soluble except if the other element is... silver, mrecury or. Ou un ester ( nitrate ) is polar what is the balanced equation for hydrogen ion... Uses in organic synthesis have been reported for example the nitration of aromatic compounds and oxidation. Chloride, however, is insoluble ( rule 3 ) are weak electrolytes O 3.. I and ammonium ions are soluble on both sides of the reaction then there is no reaction ( or... And therefore dissociated information about chromium ( III ) nitrate compounds are soluble electronegativity between the bonded atoms ( ). Other element/cation is!!!!!!!!!!!!!!! Exposure to fire or heat may result will make the unit soluble simply... 0. khanoldfriend and chlorates are soluble or not nitrate | Ba ( NO3 3. De formule chimique N O 3 − is aqueous ( aq ) because all the compounds are soluble both! Cation/Anion radii ratio and the ions ’ packing efficiency ) compounds containing these ions soluble... Is a white deliquescent solid very soluble in acetone, acetic acid, strong... - PubChem electronegativity between the bonded atoms used … more information about chromium ( III ) nitrate compounds soluble... H+ + OH- this phenomenon is termed autoionization learn with flashcards,,. Not name exceptions ) compounds containing these ions are soluble polar molecules must contain polar bonds due to,! Name exceptions ) compounds containing these ions are soluble or not ( s ): Ph.D. Biochem BS.! D'Éthyle, nitrate de cellulose, etc. solution for the compound calcium nitrate, Ca NO3... And therefore dissociated insoluble, Co ( NO3 ) 3 ) formule chimique O... Packing efficiency know if these two are soluble in water of 73.9 g/100ml at 20°C, type.


text:
# Rationalizing the Denominator

Rationalizing the denominator is pretty easy. Harder problem in rationalizing requires basic technique and knowledge in algebraic identities. In rule of math, all expressions with radical denominator are not on their simplest form. This topic is advisable if you are at least grade 8.

Application and Extension

Worked Problem 1:

Rationalize the denominator  $\displaystyle\frac{3}{\sqrt{2}}$

Solution:

$\displaystyle\frac{3}{\sqrt{2}}\cdot\displaystyle\frac{\sqrt{2}}{\sqrt{2}}$

The purpose of multiplying $\displaystyle\frac{\sqrt{2}}{\sqrt{2}}$ is to make the denominator a rational. $\displaystyle\frac{\sqrt{2}}{\sqrt{2}}=1$, multiplying 1 in the whole expression won’t change anything on the original expression.

$=\displaystyle\frac{3\sqrt{2}}{\sqrt{4}}$

$=\boxed{\displaystyle\frac{3\sqrt{2}}{2}}$

Worked Problem 2:

Simplify:  $\displaystyle\frac{5}{1-\sqrt{2}}$

Solution:

If the denominator is binomial, we need to use its conjugate. The conjugate of $2-\sqrt{2}$ is $2+\sqrt{2}$.  The conjugate of  $5-\sqrt{3}$  is  $5+\sqrt{3}$. In our given, $1+\sqrt{2}$ is the required conjugate that we are looking for.

$\displaystyle\frac{5}{1-\sqrt{2}}\cdot\displaystyle\frac{1+\sqrt{2}}{1+\sqrt{2}}$

Notice that $(1+\sqrt{2})(1-\sqrt{2})$ is in the form of $(a+b)(a-b)=a^2-b^2$, a sum and difference of two squares.

$=\displaystyle\frac{5(1+\sqrt{2})}{1^2-(\sqrt{2})^2}$

$=\boxed{\displaystyle\frac{5+5\sqrt{2}}{-3}}$

Worked Problem 3:

Rationalize the denominator:  $\displaystyle\frac{1}{\root 3\of{2}+\root 3\of{3}}$

Solution:

Recall that  $a^3+b^3=(a+b)(a^2-ab+a^2)$.

$\displaystyle\frac{1}{\root 3\of{2}+\root 3\of{3}}$

$=\displaystyle\frac{1}{\root 3\of{2}+\root 3\of{3}}\cdot\displaystyle\frac{(\root 3\of{2})^2-\root 3\of{2}\root 3\of{3}+(\root 3\of{3})^2}{(\root 3\of{2})^2-\root 3\of{2}\root 3\of{3}+(\root 3\of{3})^2}$

$=\displaystyle\frac{(\root 3\of{2})^2-\root 3\of{6}+(\root 3\of{3})^2}{(\root 3\of{2})^3+(\root 3\of{2})^3}$

$=\displaystyle\frac{(\root 3\of{2})^2-\root 3\of{6}+(\root 3\of{3})^2}{2+3}$

$=\boxed{\displaystyle\frac{\root 3\of{4}-\root 3\of{6}+\root 3\of{9}}{5}}$


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• Artificial Intelligence and Pattern Recognition •

### DG-CNN: Introducing Margin Information into Convolutional Neural Networks for Breast Cancer Diagnosis in Ultrasound Images

Xiao-Zheng Xie1 (解晓政), Jian-Wei Niu1,2 (牛建伟), Senior Member, IEEE, Xue-Feng Liu1,* (刘雪峰), Qing-Feng Li2 (李青锋), Yong Wang3 (王勇), Jie Han3 (韩洁), and Shaojie Tang4 (唐少杰), Member, IEEE

1. 1State Key Laboratory of Virtual Reality Technology and Systems, School of Computer Science and Engineering, Beihang University, Beijing 100191, China
2Beijing Advanced Innovation Center for Big Data and Brain Computing, Beihang University, Hangzhou 310051, China
3Department of Diagnostic Ultrasound, National Cancer Center, Chinese Academy of Medical Sciences, Peking Union Medical College, Beijing 100021, China
4Naveen Jindal School of Management, The University of Texas at Dallas, TX 75080-3021, U.S.A.
• Received:2019-11-26 Revised:2020-04-25 Accepted:2020-06-02 Online:2022-03-31 Published:2022-03-31
• Contact: Xue-Feng Liu E-mail:liu_xuefeng@buaa.edu.cn
• About author:Xue-Feng Liu received his M.S. and Ph.D. degrees in automatic control and aerospace engineering from the Beijing Institute of Technology, and the University of Bristol, United Kingdom, in 2003 and 2008, respectively. He was an associate professor at the School of Electronics and Information Engineering in the Huazhong University of Science and Technology, Wuhan, from 2008 to 2018. He is currently an associate professor at the School of Computer Science and Engineering, Beihang University, Beijing. His research interests include wireless sensor networks, distributed computing and in-network processing. He has served as a reviewer for several international journals/conference proceedings.
• Supported by:
This work was supported by the National Natural Science Foundation of China under Grant Nos. 61976012 and 61772060, the National Key Research and Development Program of China under Grant No. 2017YFB1301100, and China Education and Research Network Innovation Project under Grant No. NGII20170315.

Although using convolutional neural networks (CNN) for computer-aided diagnosis (CAD) has made tremendous progress in the last few years, the small medical datasets remain to be the major bottleneck in this area. To address this problem, researchers start looking for information out of the medical datasets. Previous efforts mainly leverage information from natural images via transfer learning. More recent research work focuses on integrating knowledge from medical practitioners, either letting networks resemble how practitioners are trained, how they view images, or using extra annotations. In this paper, we propose a scheme named Domain Guided-CNN (DG-CNN) to incorporate the margin information, a feature described in the consensus for radiologists to diagnose cancer in breast ultrasound (BUS) images. In DG-CNN, attention maps that highlight margin areas of tumors are first generated, and then incorporated via different approaches into the networks. We have tested the performance of DG-CNN on our own dataset (including 1485 ultrasound images) and on a public dataset. The results show that DG-CNN can be applied to different network structures like VGG and ResNet to improve their performance. For example, experimental results on our dataset show that with a certain integrating mode, the improvement of using DG-CNN over a baseline network structure ResNet18 is 2.17% in accuracy, 1.69% in sensitivity, 2.64% in specificity and 2.57% in AUC (Area Under Curve). To the best of our knowledge, this is the first time that the margin information is utilized to improve the performance of deep neural networks in diagnosing breast cancer in BUS images.

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[9] Tajbakhsh N, Shin J Y, Gurudu S R, Hurst R T, Kendall C B, Gotway M B, Liang J. Convolutional neural networks for medical image analysis: Full training or fine tuning? IEEE Trans. Med. Imaging, 2016, 35(5): 1299-1312. DOI: 10.1109/TMI.2016.2535302. [10] Guan Q, Huang Y, Zhong Z, Zheng Z, Zheng L, Yang Y. Diagnose like a radiologist: Attention guided convolutional neural network for thorax disease classification. arXiv:1801.09927, 2018. https://arxiv.org/abs/1801.09927, Nov. 2021. [11] González-Díaz I. DermaKNet: Incorporating the knowledge of dermatologists to convolutional neural networks for skin lesion diagnosis. IEEE J. Biomed. Health Inform., 2018, 23(2): 547-559. DOI: 10.1109/JBHI.2018.2806962. [12] Li L, Xu M, Wang X, Jiang L, Liu H. Attention based glaucoma detection: A large-scale database and CNN model. In Proc. the IEEE Conference on Computer Vision and Pattern Recognition, June 2019, pp.10571-10580. DOI: 10.1109/CVPR.2019.01082. [13] Fang L, Wang C, Li S, Rabbani H, Chen X, Liu Z. Attention to lesion: Lesion-aware convolutional neural network for retinal optical coherence tomography image classification. IEEE Trans. Med. Imaging, 2019, 38(8): 1959-1970. DOI: 10.1109/TMI.2019.2898414. [14] Mitsuhara M, Fukui H, Sakashita Y, Ogata T, Hirakawa T, Yamashita T, Fujiyoshi H. Embedding human knowledge in deep neural network via attention map. arXiv:1905.03540, 2019. https://arxiv.org/abs/1905.03540, May 2021. [15] Dorsi C, Bassett L, Feisg S, Lee C I, Lehman C D, Bassett L W. Breast Imaging Reporting and Data System (BI-RADS). Oxford University Press, 2018. [16] Bian C, Lee R, Chou Y, Cheng J. Boundary regularized convolutional neural network for layer parsing of breast anatomy in automated whole breast ultrasound. In Proc. the 20th International Conference on Medical Image Computing and Computer-Assisted Intervention, Sept. 2017, pp.259-266. DOI: 10.1007/978-3-319-66179-7\textunderscore 30. [17] Maicas G, Bradley A P, Nascimento J C, Reid I, Carneiro G. Training medical image analysis systems like radiologists. In Proc. the 21st International Conference on Medical Image Computing and Computer-Assisted Intervention, September 2018, pp.546-554. DOI: 10.1007/978-3-030-00928-1\textunderscore 62. [18] Liu J, Li W, Zhao N, Cao K, Yin Y, Song Q, Chen H, Gong X. Integrate domain knowledge in training CNN for ultrasonography breast cancer diagnosis. In Proc. the 21st International Conference on Medical Image Computing and Computer-Assisted Intervention, September 2018, pp.868-875. DOI: 10.1007/978-3-030-00934-2\textunderscore 96. [19] Wang X, Peng Y, Lu Y, Lu Z, Summers R M. TieNet: Text-image embedding network for common thorax disease classification and reporting in chest X-rays. In Proc. the 2018 IEEE/CVF Conference on Computer Vision and Pattern Recognition, June 2018, pp.9049-9058. DOI: 10.1109/CVPR.2018.00943. [20] Berg W A, Cosgrove D O, Doré C J et al. Shear-wave elastography improves the specificity of breast US: the BE1 multinational study of 939 masses. Radiology, 2012, 262(2): 435-449. DOI: 10.1148/radiol.11110640. [21] Dobruch-Sobczak K, Piotrzkowska-Wróblewska H, Roszkowska-Purska K, Nowicki A, Jakubowsi W. Usefulness of combined BI-RADS analysis and Nakagami statistics of ultrasound echoes in the diagnosis of breast lesions. Clin. Radiol., 2017, 72(4): 339-339. DOI: 10.1016/j.crad.2016.11.009. [22] Liu Y, Cheng M M, Hu X, Wang K, Bai X. Richer convolutional features for edge detection. In Proc. the IEEE Conference on Computer Vision and Pattern Recognition, July 2017, pp.5872-5881. DOI: 10.1109/CVPR.2017.622. [23] Arbeláez P, Maire M, Fowlkes C, Malik J. Contour detection and hierarchical image segmentation. IEEE Trans. Pattern Anal. Mach. Intell., 2011, 33(5): 898-916. DOI: 10.1109/TPAMI.2010.161. [24] Lin M, Chen Q, Yan S. Network in network. arXiv:1312.4400, 2013. https://arxiv.org/abs/1312.4400, March 2021. [25] Zhou B, Khosla A, Lapedriza A, Oliva A, Torralba A. Learning deep features for discriminative localization. In Proc. the IEEE Conference on Computer Vision and Pattern Recognition, June 2016, pp.2921-2929. DOI: 10.1109/CVPR.2016.319. [26] Shelhamer E, Long J, Darrell T. Fully convolutional networks for semantic segmentation. IEEE Trans. Pattern Anal. Mach. Intell., 2017, 39(4): 640-651. DOI: 10.1109/TPAMI.2016.2572683. [27] He K, Gkioxari G, Dollar P, Girshick R. Mask R-CNN. In Proc. the IEEE International Conference on Computer Vision, October 2017, pp.2980-2988. DOI: 10.1109/ICCV.2017.322. [28] Han S, Kang H K, Jeong J Y et al. A deep learning framework for supporting the classification of breast lesions in ultrasound images. Phys. Med. Biol., 2017, 62(19): 7714-7728. DOI: 10.1088/1361-6560/aa82ec. [29] Ren S, He K, Girshick R, Sun J. Faster R-CNN: Towards real-time object detection with region proposal networks. IEEE Trans. Pattern Anal. Mach. Intell., 2017, 39(6): 1137-1149. DOI: 10.1109/TPAMI.2016.2577031.
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# 2.1: Conditional Probability

Difficulty Level: At Grade Created by: CK-12

Learning Objectives

• Know the definition of conditional probability.
• Use conditional probability to solve for probabilities in finite sample spaces.

INDEPENDENT EVENTS – Outcomes of events are not affected by other events (in other words – random events).

DEPENDENT EVENTS – The outcome of one event is affected by another event.

MUTUALLY EXCLUSIVE EVENTS – When two events cannot occur at the same time (in a single roll, rolling a 3 on a die and rolling an even number on a die are mutually exclusive).

MUTUALLY INCLUSIVE EVENTS – When two events can occur at the same time (in a single roll, rolling a 3 on a die and rolling an odd number on a die are mutually exclusive).

In the previous section we looked at probability in terms of events that are independent and dependent, mutually inclusive and mutually exclusive. Take a look in the box to your left just to recall the definitions of these terms.

The next type of event probability is called CONDITIONAL PROBABILITY. With conditional probability, the probability of the second event DEPENDS ON the probability of the first event.

Conditional Probability

P(BA)P(AB)=P(AB)P(A)=P(A)×P(BA)

Another way to look at the conditional probability formula is:

P(secondfirst)=P(first choice and second choice)P(first choice)

ABC\begin{align*}ABC\end{align*} High School students are required to write an entrance test to the statistics course before beginning the course. The following table represents the data collected regarding this year’s group. The numbers represent the number of students in each group.

Studied Not Studied
Passed 17 3
Not Passed 2 23

Questions

1. Discover the following probabilities:

a. P\begin{align*}P\end{align*}(pass and studied)

b. P\begin{align*}P\end{align*}(studied) and

c. P\begin{align*}P\end{align*}(pass/studied)

Remember when you have completed this unit you will be see this problem again to solve it.

Let’s work through a few examples of conditional probability to see how the formula works.

Example 1: A bag contains green balls and yellow balls. You are going to choose two balls without replacement. If the probability of selecting a green ball and a yellow ball is 1439\begin{align*}\frac{14}{39}\end{align*}, what is the probability of selecting a yellow ball on the second draw, if you know that the probability of selecting a green ball on the first draw is 49\begin{align*}\frac{4}{9}\end{align*}.

Solution:

Step 1: List what you know

P(Green)P(Green AND Yellow)=49=1439

Step 2: Calculate the probability of selecting a yellow ball on the second draw with a green ball on the first draw

P(Y|G)P(Y|G)P(Y|G)P(Y|G)P(Y|G)=P(Green AND Yellow)P(Green)=143949=1439×94=126156=2126

Step 3: Write your conclusion: Therefore the probability of selecting a yellow ball on the second draw after drawing a green ball on the first draw is 2126\begin{align*}\frac{21}{26}\end{align*}.

Example 2: Music and Math are said to be two subjects that are closely related in the way the students think as they learn. At the local high school, the probability that a student takes math and music is 0.25. The probability that a student is taking math is 0.85. What is the probability that a student that is in music is also choosing math?

Solution:

Step 1: List what you know

P(Math)P(Math AND Music)=0.85=0.25

Step 2: Calculate the probability of choosing music as a second course when math is chosen as a first course.

P(Music|Math)P(Music|Math)P(Music|Math)P(Music|Math)=P(Math AND Music)P(Math)=0.250.85=0.29=29%

Step 3: Write your conclusion: Therefore, the probability of selecting music as a second course when math is chosen as a first course is 29%.

Example 3: The probability that it is Friday and that a student is absent is 0.05. Since there are 5 school days in a week, the probability that it is Friday is 15\begin{align*}\frac{1}{5}\end{align*} or 0.2. What is the probability that a student is absent given that today is Friday?

Solution:

Step 1: List what you know

P(Friday)P(Friday AND Absent)=0.20=0.05

Step 2: Calculate the probability of being absent from school as a second choice when Friday is chosen as a first choice.

P(Absent|Friday)P(Absent|Friday)P(Absent|Friday)P(Absent|Friday)=P(Friday AND Absent)P(Friday)=0.050.20=0.25=25%

Step 3: Write your conclusion: Therefore the probability of being absent from school as a second choice when the day, Friday, is chosen as a first choice is 25%.

Example 4: Students were asked to use computer simulations to help them in their studying of mathematics. After a trial period, the students were surveyed to see if the technology helped them study or did not. A control group was not allowed to use technology. They used a textbook only. The following table represents the data collected regarding this group. The numbers represent the number of students in each group.

Technology Textbooks
Improved studying 25 2
Did not improve studying 3 30

Discover the following probabilities:

a. P(Improved studying and used technology)

b. P(Improved studying and

c. P(Improved studying/used technology)

Solution:

Total students =25+2+3+30=60\begin{align*}= 25 + 2 + 3 + 30 = 60\end{align*}

a. P\begin{align*}P\end{align*}(Improved studying and used technology) =2560\begin{align*}=\frac{25}{60}\end{align*}

P\begin{align*}P\end{align*}(Improved studying and used technology) =2560\begin{align*}=\frac{25}{60}\end{align*}

b. P\begin{align*}P\end{align*}(Improved studying) =2560+260\begin{align*}=\frac{25}{60} + \frac{2}{60}\end{align*}

P\begin{align*}P\end{align*}(Improved studying) =2760\begin{align*}=\frac{27}{60}\end{align*}

c. P(Improved studying|used technology)=P(used technology AND improved studying)P(used technology)\begin{align*}P(Improved\ studying | used\ technology) = \frac{P(used\ technology\ AND\ improved\ studying)}{P(used\ technology)}\end{align*}

P(Improved studying|used technology)P(Improved studying|used technology)P(Improved studying|used technology)P(Improved studying|used technology)=25602860=2560×6028=2528=89%

Therefore the probability of improving studying when choosing technology was 89%.

Now let’s go back to our original problem from the beginning of this chapter.

ABC\begin{align*}ABC\end{align*} High School students are required to write an entrance test to the statistics course before beginning the course. The following table represents the data collected regarding this year’s group. The numbers represent the number of students in each group.

Studied Not Studied
Passed 17 3
Not Passed 2 23

Questions

2. Discover the following probabilities:

a. P(pass and studied)

b. P(studied, and

c. P(pass/studied)

Solution:

Total students =17+3+2+23=45\begin{align*}= 17 + 3 + 2 + 23 = 45\end{align*}

a. P\begin{align*}P\end{align*}(passed and studied) =1745\begin{align*}=\frac{17}{45}\end{align*}

P\begin{align*}P\end{align*}(Improved studying and used technology) =2560\begin{align*}=\frac{25}{60}\end{align*}

b. P\begin{align*}P\end{align*}(studied) =1745+245\begin{align*}=\frac{17}{45} + \frac{2}{45}\end{align*}

P\begin{align*}P\end{align*}(studied) =1945\begin{align*}=\frac{19}{45}\end{align*}

c. P(passed|studied)=P(studied AND passed)P(studied)\begin{align*}P(passed | studied) = \frac{P(studied\ AND\ passed)}{P(studied)}\end{align*}

P(passed|studied)P(passed|studied)P(passed|studied)P(passed|studied)=17451945=1745×4519=1719=89%

Therefore the probability of passing the course when studying was 89%.

Lesson Summary

The lesson was an extension of the previous chapter on probability. Here we learned about conditional probability or probability of events where the probability of the second occurrence is dependent on the probability of the first event. In other words, it is a probability calculation where conditions have been into place. No longer can you simply pick cards and find the probability, for example, you will now be told that the choosing of the cards have conditions. Conditions such as the first card must be a heart.

Points to Consider

• How is the conditional formula related to the previous probability formulas learned?
• Are tables a good way to visualize probability?

Vocabulary

Conditional Probability
The probability of a particular dependent event, given the outcome of the event on which it depends.

Review Questions

1. A card is chosen at random. What is the probability that the card is black and is a 7?
2. A card is chosen at random. What is the probability that the card is red and is a jack of spades?
3. A bag contains 5 blue balls and 3 pink balls. Two balls are chosen at random and not replaced. What is the probability of choosing a blue ball after choosing a pink ball?
4. Kaj is tossing two coins. What is the probability that he will toss 2 tails given that the first toss was a tail?
5. A bag contains blue balls and red balls. You are going to choose two balls without replacement. If the probability of selecting a blue ball and a red ball is \begin{align*}\frac{13}{42}\end{align*}, what is the probability of selecting a red ball on the second draw, if you know that the probability of selecting a blue ball on the first draw is \begin{align*}\frac{7}{13}\end{align*}.
6. In a recent survey, 100 students were asked to see whether they would prefer to drive to school or bike. The following data was collected.
Drive Bike
Male 28 14
Female 18 40

(a) Find the probability that the person surveyed would want to drive, given that they are female.

(b) Find the probability that the person surveyed would be male, given that they would want to bike to school.

1. The little league baseball team is open to both boys and girls. The probability that a person joining the little league team and being a girl is 0.265. Of the 386 possible youth in the town to play little league ball, only 157 are girls, or 40.7%. What is the probability that a youth joining the league will be a girl?

1. \begin{align*}\frac{1}{13}\end{align*}
2. 0
3. \begin{align*}\frac{5}{7}\end{align*}
4. \begin{align*}\frac{1}{3}\end{align*}
5. \begin{align*}\frac{169}{294}\end{align*}
1. \begin{align*}\frac{9}{29}\end{align*}
2. \begin{align*}\frac{7}{27}\end{align*}
6. \begin{align*}\frac{265}{407}\end{align*}

Answer Key for Review Questions (even numbers)

2. 0

4. \begin{align*}\frac{1}{3}\end{align*}

6. (a) \begin{align*}\frac{9}{29}\end{align*}

(b) \begin{align*}\frac{7}{27}\end{align*}

## Date Created:

Feb 23, 2012

Dec 29, 2014
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# RTC circuit supercapacitor charge/discharge doubts

This is my current RTC circuit...

Supercapacitor = DXJ-5R5V334U (ELNA)... DXJ-5R5V334U = 0.33F, 5.5V, ESR = 150 Ohms Resistor = 150 ohms 0603 100mW 1%

If I consider only the resistor, I have a RC constant of 0.33 * 150 = 49.5 (~50) seconds... 50 seconds x 3 = 150 seconds (3x RC = capacitor very well charged, 95%...). That is 150 seconds... divided by 60 = 2,5 minutes...

But if I consider also the ESR of the supercapacitor, I will have 300 Ohms in total, so that the charging time would be 5 minutes, double the time...

What value of resistance should I consider in practice? 150 ohms (2.5 minutes)? Or 300 ohms (5 minutes)?

Another question related to that, is: the VBAT pin consumption is 1.2uA (@ 3.3V)... If the supercapacitor is fully charged (like 95%) and the circuit loss the power supply of 3.3V (shutdown/off), the RTC will continue to operate from the VBAT pin (supercapacitor), so how much time the RTC would stay working from VBAT? (1 month? more?). As I said, consumption is 1.2uA (@ 3.3V) and the capacitor is 0.33F charged with something like 3 to 3.1V, how much time the RTC IC would stay working if the 3.3V rail keeps off "foreve"r? According the the datasheet of RTC chip, the "Backup supply voltage" minimum voltage is 1.3V. Do you know some online calculator where I could estimate that time of operating from VBAT pin after power-off of the board and capacitor being 95% charged? Could you calculate these things to me please? Or maybe an online calculator would be great also...

Thanks and Best Regards...

I use this website all the time to quickly calculate exactly what you're asking. https://sparks.gogo.co.nz/capacitor-formulae.html

• Should I consider 150 ohms or 300 ohms? that's the main doubt... Oct 15, 2020 at 16:02
• 150 ohms should be fine for just about any regulator. When the capacitor is at 0v, you will have an initial current of 22mA. That will continue to decrease the more the capacitor is charged. I would figure out how much current your regulator is able to handle. If its able to handle 1A, I'd use a 10ohm resistor which will give you a max current of 330mA and will charge the capacitor in about 10 seconds. With a 47ohm your max initial draw will be 70mA and charge time will be 46 seconds. Oct 15, 2020 at 19:06
• thanks mate. My doubt is if the ESR of the supercap should be considered or not to calculate the charge current... for example, it has 150 ohms of ESR, so it would be not needed to add any extra resistor in series like my schematic shows, because that 150 ohms of ESR of the supercap it would limit the current in 22mA just by its ESR... Got? Oct 15, 2020 at 20:09
• That is correct. Thats a pretty high ESR though. Oct 15, 2020 at 20:32

What value of resistance should I consider in practice? 150 ohms (2.5 minutes)? Or 300 ohms (5 minutes)?

The ESR of the cap plus external resistance. You should model the cap as a "pure" capacitance with a 150$$\\Omega\$$ resistor in series.

Do you know some online calculator where I could estimate that time of operating from VBAT pin after power-off of the board and capacitor being 95% charged?

Well this is on line, presumably you have a calculator, so bear with me a moment...

The Math

$$\V_{cap} = \frac{q_c}{C}\$$, where $$\q\$$ is the charge on the cap in coulombs, and $$\C\$$ is the capacitance. And $$\q_c(t) = \int i_c dt\$$. It's probably safe to assume that the RTC clock current consumption is constant or goes down as voltage goes down, but your should check this. If true, then the change in charge is just equal to current times time: $$\\Delta q_c = \Delta i_c t\$$. Then $$\\Delta V_c = \frac{\Delta q_c}{C} = \frac{i_c \Delta t}{C}\$$.

If you want to go from 3.1V down to 1.3V, then $$\\Delta V_c = \mathrm{3.1V - 1.3V = 1.8V}\$$. Then doing a bit of algebra gets $$\\Delta t = \frac{C \Delta V_c}{i_c}\$$. Using your numbers, $$\\Delta t\$$ is then $$\\mathrm{\frac{1.8V\ 0.33F}{1.2\mu A} = 495 \cdot 10^3}\$$ seconds.

However -- Shotkey diodes leak, particularly at high temperatures. You may want to revisit your diode choice.


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# the sum of the first 8 terms of a geometric sequence is 6560. The common ratio is 3. What is the third term of the sequence?

the sum of the first 8 terms of a geometric sequence is 6560. The common ratio is 3. What is the third term of the sequence?
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hesgidiauE

The formula for the sum of nn terms of a geometric sequence is ${S}_{n}=\frac{{a}_{1}\left(1-{r}^{n}\right)}{\left(1-r\right)}$ where ${a}_{1}$ is the first term and rr is the common ratio.
If the sum of the first $n=8$ terms is ${S}_{8}=6560$ and the common ratio is $r=3$, then:
${S}_{8}=\frac{{a}_{1}\left(1-{3}^{8}\right)}{1-3}$
$6560=\frac{{a}_{1}\left(1-6561\right)}{-2}$
$6560=\frac{\frac{-6560{a}_{1}}{-2}}{}$
$6560=-3280{a}_{1}$
$-2={a}_{1}$
Now that we know ${a}_{1}$, we can find the third term. The nnth term of a geometric sequence is given by the formula ${a}_{n}={a}_{1}{r}^{n}-1$.
Since ${a}_{1}=-2,r=3,$ and $n=3$ for the third term, then:
${a}_{3}=-2{\left(3\right)}^{3}-1=-2{\left(3\right)}^{2}=-2\left(9\right)=-18$


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# Solvation Energies

Solvation energies are usually decomposed into a free energy cycle as shown in the free energy cycle below. Note that such solvation energies often performed on fixed conformations; as such, they are more correctly called “potentials of mean force”. More details on using APBS for the polar and nonpolar portions of such a cycle are given in the following sections.

### Solvation free energy cycle

Our model solvation free energy cycle is shown here

This cycle incorporates several processes into the solvation energy (step 1). Step 2 indicates charging of the solute in solution (e.g., inhomogeneous dielectric, ions present). Step 3 indicates the introduction of attractive solute-solvent dispersive interaction interactions (e.g., an integral of Weeks-Chandler-Andersen interactions over the solvent-accessible volume). Step 4 indicates the introduction of repulsive solute-solvent interaction (e.g., cavity formation). Steps 5 and 6 are basically null steps although they could be used to offset unwanted energies added in Steps 3 and 4 above. Finally, Step 6 represents the charging of the solute in a vacuum or homogeneous dielectric environment in the absence of mobile ions.

### Polar solvation

The full free energy cycle is usually decomposed into polar and nonpolar parts. The polar portion is usually represented by the charging energies in Steps 2 and 6:

$\Delta_p G = \Delta_2 G - \Delta_6 G$

Energies returned from APBS electrostatics calculations are charging free energies. Therefore, to calculate the polar contribution to the solvation free energy, we simply need to setup two calculations corresponding to Steps 2 and 6 in the free energy cycle. Note that the electrostatic charging free energies returned by APBS include self-interaction terms. These are the energies of a charge distribution interacting with itself. Such self-interaction energies are typically very large and extremely sensitive to the problem discretization (grid spacing, location, etc.). Therefore, it is very important that the two calculations in Steps 2 and 6 are performed with identical grid spacings, lengths, and centers, in order to ensure appropriate matching (or “cancellation”) of self-energy terms.

### Apolar solvation

Referring back to the solvation free energy cycle, the nonpolar solvation free energy is usually represented by the energy changes in Steps 3 through 5:

$\Delta_n G = (\Delta_3 G - \Delta_5 G) + \Delta_4 G$

where Step 4 represents the energy of creating a cavity in solution and Steps 3-5 is the energy associated with dispersive interactions between the solute and solvent. There are many possible choices for modeling this nonpolar solvation process. APBS implements a relatively general model described by Wagoner and Baker (PNAS 2006) and references therein. The implementation and invocation of this model is described in more detail in the APBS user guide. Our basic model for the cavity creation term (Step 4) is motivated by scaled particle theory and has the form

$\Delta_4 G = pV + \gamma A$

where is the solvent pressure (press keyword in the APOLAR input file section), V is the solute volume, $\gamma$ is the solvent surface tension (gamma keyword in the APOLAR input file section), and A is the solute surface area.

Our basic model for the dispersion terms (Steps 3 and 5) follow a Weeks-Chandler-Anderson framework as proposed by Levy, Zhang, and Gallicchio (J Comput Chem, 2002):

$\Delta_3 G - \Delta_5 G = \overset{-} \rho \int_\omega u^{(att)}(y)\theta(y)dy$

where $\overline{\rho}$ is the bulk solvent density (bconc keyword in the APOLAR input file section), $\Omega$ is the problem domain, $u^{\mathrm{(att)}}(y)$ is the attractive dispersion interaction between the solute and the solvent at point y with dispersive Lennard-Jones parameters specified in APBS parameter files, and $\theta(y)$ describes the solvent accessibility of point y.

The ability to independently adjust press, gamma, and bconc means that the general nonpolar solvation model presented above can be easily adapted to other popular nonpolar solvation models. For example, setting press and bconc to zero yields a typical solvent-accessible surface area model.

Unlike the Born ion, there are no good simple examples for demonstrating these types of nonpolar calculations. APBS includes several examples of calculations using the apolar model above. Interested readers should examine the alkanes apolar examples provided with APBS.

### Examples

Additional solvation energy calculations are provided below:

The Born ion

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# One-time pad encryption - how does the receiving side decrypt the message?

If a message is sent using one-time pad scheme, which generates random bits in order to, for example, XOR them with the message's bits - how does the receiving side generate the exact series of bits ?

If the receiving side is able to generate the same series then how is it still random ?

• The receiving side needs to receive an explicit copy of the random bits. – SEJPM Feb 9 '19 at 14:45
• And said random bits have to remain secret. In the rare practical cases when the OTP is used, the random bits are generated in advance of transmission, and sent by secure means such as trusted couriers. – fgrieu Feb 9 '19 at 15:18

## 1 Answer

The one-time pad requires pre-shared key. Usually, one of the two parties generates the keystream, write into a paper or CD/DVD, or USB etc.. and sends the keystream to the other party in a secure way. The secure way is not encryption, it is using some men to carry it to the target. After the transmission, they can use the key to decipher.

Wikipedia lists historical uses of one-time pad. One case is very important; the Venona Project. At some point, the Russian had reused some part of the key. So, the question is how one notices this? The USA and British Code-breakers always tried the crib-dragging and one day were successful!

Some of The Crib Draggin answers in CryptoSO;

• A coincidence test suffices to detect re-use. No need for crib-dragging for that. See the classic texts by Friedman etc. – Henno Brandsma Feb 10 '19 at 22:19
• @HennoBrandsma good point. When I see the coincidence test I remembered but needs to refresh. Thanks. – kelalaka Feb 10 '19 at 22:31


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# Solutions of the equation $2^{q-1} \equiv q \pmod {4q^2+1 }$ where $q$ is an odd prime

I am interested in the solutions of the equation $2^{q-1} \equiv q \pmod {p}$ where $p=4q^2+1$ for an odd prime $q$.

So far the only solution I found by trial and error is $q=193$ but I don't know how to realize it as a solution. I'd like to know if there are other solutions.

By relaxing the condition on $q$ one gets two more solutions: $q=1,2$. It would also be interesting to see if there are other positive integer solutions.

Of the $148932$ odd primes $q<2000000$, the only solution to your equation is $q=193$. If you drop the primality restriction on $q$, then as well as $q=1$ and $2$, $q=29570$ is also a solution (with, as it happens, $p$ prime).
I've checked all primes $q$ below $10^{11}$ with no new solutions found (even if the primality of $p=4q^2+1$ is relaxed).
Notice that $2^{q-1}\equiv q\pmod{4q^2+1}$ implies $2^{2q}\equiv 4q^2\equiv -1\pmod{4q^2+1}$, further implying that the multiplicative order of $2$ is $4q$ (also, 2 is a $q$-th power residue) modulo $4q^2+1$. Heuristically, this has probability $\frac{1}{q}$ and since primes $4q^2+1$ with prime $q$ are rather sparse (cf. A052292 in the OEIS), this likely means that there are only a finite number of suitable $q$.